ࡱ> 'EG 89:;<=>?@ABCDIJ"<  } C " # YZtHm  }bjbj _ hhj1,,oo8 t \4!)))-ppp p]______2ԟp_Yq8h8ppqq_oo)-:]~]~]~qo)-]]~q]]~]~)'-UrZIΛ0DyD4Dp"p]~ppppp__}pppqqqqDppppppppp, 5: Section 1: The Tangent and Velocity Problems SOLs: APC.12: The student will apply the derivative to solve problems, including tangent and normal lines to a curve, curve sketching, velocity, acceleration, related rates of change, Newton's method, differentials and linear approximations, and optimization problems. Objectives: Students will be able to: Understand the tangent problem Understand the velocity problem Vocabulary: Tangent a line that touches the curve in only one point; same slopes at intersection point Secant a line that intersects the curve in only two points; average slope between two points Average Velocity distance traveled divided by time elapsed average of two instantaneous velocities; the slope of the secant line Instantaneous Velocity the velocity at an instant in time; limit of average velocity as the two points of the secant get closer together (as "x !0) the slope of the tangent line Key Concepts: Tangent Problem: Slope of a curve at any point (slope of a tangent at that point) can be estimated by taking the slope of the secant line with those two x-values being very close together Velocity Problem: Instantaneous velocity is the limit of the average velocities as the change in time approaches 0 (as "t !0) A secant line at P(a,b) for the graph of y = f(x) is a line joining P and another point Q also on the graph. If Q has coordinates (c, d) then the slope of the secant line is "y / "x = (d - b) / (c - a). A tangent line to the graph of a function touches the graph at a given point. When we are given discrete data and asked to find the equation of a tangent line, we must use the secant line as an estimator. If f(x) is interpreted as the distance an object is located from the origin along an x-axis at time x, then Slope of the secant line from P to Q is the average velocity from P to Q Slope of the tangent line at P is the instantaneous velocity. We can find instantaneous velocity using the derivative. Example: The distance a runner is from the starting line is given in the table t (seconds)01234d (meters)0491624 a. Find the average velocity over the time intervals i. [1, 4] ii. [1, 3] iii. [1, 2] iv. [0, 1] b. Estimate the instantaneous velocity at t = 1.  EMBED PowerPoint.Show.8   EMBED PowerPoint.Show.8  Concept Summary: The slope of the secant approaches the slope of the tangent at a point on the curve as "x !0 Homework  Problems: pg 91-92: 1, 5, 8 Read: Section 2.2 Skills for Next Section: Limits Section 2: The Limit of a Function SOLs: APC.2: The student will define and apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at infinity, infinite limits, and nonexistent limits. Objectives: Students will be able to: Determine and Understand one-sided limits Determine and Understand two-sided limits Vocabulary: Limit (two sided) as x approaches a value a, f(x) approaches a value L Left-hand (side) Limit as x approaches a value a from the negative side, f(x) approaches a value L Right-hand (side) Limit as x approaches a value a from the positive side, f(x) approaches a value L DNE  does not exist (either a limit increase/decreases without bound or the two one-sided limits are not equal) Infinity  increases (+") without bound or decreases (-") without bound [NOT a number!!] Vertical Asymptote at x = a because a limit as x approaches a either increases or decreases without bound Key Concepts:  EMBED PowerPoint.Show.8   EMBED PowerPoint.Show.8  Remember: around a point where a function does not exist, trying to guess the limit may have some pitfalls as examples 2 through 6 go through in the book. A calculator is not a replacement for a thinking mind!  means that as x gets closer and closer to the number a (but not equal to a), the corresponding f(x) values get closer and closer to the number L. Example 1: Answer each using the graph to the right (from Study Guide that accompanies Single Variable Calculus by Stewart)  a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.   2. Use tables to estimate 3. Use algebra to find: a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Look at page 94-95, ex 4 & 5 for two tricky limits.  , the left-hand limit of f at a means that f(x) gets closer and closer to L as x approaches a and x < a. It is called left-hand because it only concerns values of x less than a (to the left of a on the x-axis)  , the right-hand limit is a similar concept but involves only values of x greater than a. If and both exist and are the same number L, then exists and equals L. If either one sided limit does not exist or, or they exist but are different numbers, then does not exist. If , then both one sided limits exist and are equal to L. (see page 98 for diagrams)  Examples: 1. Use the graph to answer each of the following: a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  e.  EMBED Equation.3  f.  EMBED Equation.3  g. EMBED Equation.3  h. EMBED Equation.3  i.  EMBED Equation.3  2.   3. Find, where 4. Always, sometimes, or never true:  a. If does not exist, then does not exist.  b. If does not exist, then does not exist. Sometimes does not exist because as x is assigned values that approach a the corresponding f(x) values grow larger without bound. In this case we say:  Remember: still means the limit does not exist, but it fails to exist in this special way we get a direction! The graph goes up without bound. means f(x) becomes smaller without bound as x approaches a, but the graph goes off in the opposite direction. Example: Evaluate the limits given the graph of f(x) : a.  EMBED Equation.3  f.  EMBED Equation.3  b. EMBED Equation.3  g. EMBED Equation.3  c.  EMBED Equation.3  h. EMBED Equation.3  d.  EMBED Equation.3  i.  EMBED Equation.3  e.  EMBED Equation.3   True/False: If and , then . Homework Problems: pg 102-104: [Day 1] 5, 6, 7, 9; [Day 2] 12, 19, 21, 23, 24, 27 Read: Section 2.3 Skills for Next Section: Laws of Limits Section 3: Calculating Limits Using the Limit Laws SOLs: APC.2: The student will define and apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at infinity, infinite limits, and nonexistent limits. Objectives: Students will be able to: Find Limits using the laws of Limits Understand and use the Squeeze Theorem Vocabulary: Greatest Integer Function [[ x ]], the largest integer that is less than or equal to x Continuous (studied in detail in 2.5) no interrupt or abrupt change in the function Key Concept:  EMBED PowerPoint.Show.8   EMBED PowerPoint.Show.8  Examples: Find each of the following  1.  EMBED Equation.3  2.  3.  EMBED Equation.3  4.  EMBED Equation.3  where    5. given  6.  7. If a function g(x) is trapped or sandwiched or squeezed between two functions f(x) and h(x) with a known limit L as x approaches c, then the limit of g(x) will also be L. This is called the squeeze theorem look on page 110 for the formal definition and page 111 for the picture. This theorem allows us to find limits of strange functions and to prove some limits analytically.  Prove: Homework Problems: pg 111-113: [Day 1] 1, 3, 6, 10, 11, 13, 20 [Day 2] 33, 40, 41, 52 Read: Section 2.4 Section 2.4: The Precise Definition of a Limit SOLs: APC.2: The student will define and apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at infinity, infinite limits, and nonexistent limits. Objectives: Students will be able to: Define and use the precise definition of a limit Vocabulary: Error tolerances   closeness as x approaches a   epsilon, associated with distances about f(x)   delta, associated with distances about x Key Concept:  EMBED PowerPoint.Show.8   Intuitive meaning: To say means that when x is very close to a (but not necessarily equal to a), f(x) is near L.  Consider  EMBED Equation.3 . From the graph and table, we can see that the limit is 3. x1.251.11.011.0011.999.99.9f(x)3.8133.313.033.003undef2.9972.9702.71 We can also find the limit analytically by factoring, simplifying and substituting:  EMBED Equation.3   Precise definition: you must memorize this! Look at pages 115-116 Let f be a function on some open interval that contains the number a, except possibly a itself. Then we say that the limit of f(x) as x approaches a is L and we write IF for every number e > 0 there is a corresponding d >0 such that | f(x)  L| < e whenever 0 < |x  a| < d This means that given a certain e, a certain precision or tolerance that we seek; we can find a corresponding d. In other words, in order to get within a certain precision of the limit, how close must we be to a? Example: Given that  EMBED Equation.3 , find d such that e < .01. Solution: Identify f(x), L, a and fill in the definition | f(x)  L| < e provided 0 < |x  a| < d |(3x  7)  5| < 0.01 ( |3x 12| < 0.01 ( 3|x 4| < 0.01 ( |x 4| < 0.0033 This means in order to guarantee that we are within .01 of the limit, we must start within .003 of a = 4 Look at page 123: #1, 4, 5, 6  A proof that consists of a verification of the definition. Start by assuming you have e > 0 and then determine a number d so that the statement | f(x)  L| < e can be deduced from the statement 0 < |x  a| < d. So, your proof consists of 1. assuming e > 0 2. stating your choice for d 3. showing that 0 < |x  a| < d implies | f(x)  L| < e Example: Prove  For e > 0, choose Then, for 0 < |x  3| <   EMBED Equation.3   In summary, if 0 < |x  3| <  then |(6x  5)  13|<  thus . There are similar precise definitions for limits at " and -" (look at pages 121-122). You must be able to recognize them though you do not have to memorize them as you do the first definition. Homework Problems: pg 122-124: 6, 15 Read: Section 2.5 Section 2.5: Continuity SOLs: APC.3: The student will state the definition of continuity and determine where a function is continuous or discontinuous. This will include continuity at a point; continuity over a closed interval; application of the Intermediate Value Theorem; and graphical interpretation of continuity and discontinuity. Objectives: Students will be able to: Understand and use the definition of continuity Understand and use the Intermediate Value Theorem Vocabulary: Continuity no gaps in the curve (laymans definition) Discontinuity a point where the function is not continuous Removable discontinuity a discontinuity that can be removed by redefining the function at a point also called a point discontinuity Infinite discontinuity a discontinuity because the function increases or decreases without bound at a point Jump discontinuity a discontinuity because the function jumps from one value to another Continuous from the right at a number a the limit of f(x) as x approaches a from the right is f(a) Continuous from the left at a number a the limit of f(x) as x approaches a from the left is f(a) A function is continuous on an interval if it is continuous at every number in the interval Key Concept:  EMBED PowerPoint.Show.8  Examples: 1. Is continuous at x = 1? 2. Is f(x) = x + 1 continuous at x = 0?  3. Is continuous at x = 0?  4. Is continuous at x = 2?  5. Is continuous at x = -1? At x = 1? Theorem: A polynomial function is continuous at every real number a. A rational function is continuous at every real number a in its domain, i.e., except where its denominator is zero. The absolute value function is continuous at every real number a. If n is odd, the nth root function is continuous at every real number a, if n is even, it is continuous at every positive real number a. (page 127) If f and g are continuous at a, then so are:   Composite Limit Theorem: If and if f is continuous at L, then In particular, if g is continuous at c and f is continuous at g(c), then the composite (f % g) is continuous at c.  Example: Is continuous for all real numbers? Continuity on an interval: Definition: We say f is continuous on an open interval if it is continuous at each point on that interval. We say f is continuous on the closed interval [a, b] if it is continuous on (a, b), right continuous at a, and left continuous at b. (p 124)  Example 1: f(x) = [x] is continuous on (1, 2);  is continuous on [1, 2]. Example 2: a) At which points is the graph discontinuous? b) On what intervals is the graph continuous? Intermediate Value Theorem: If f is continuous on [a, b] and if W is a number between f(a) and f(b), then there is a number c between a and b such that f(c) = W. (page 129) Example: Show that f(x) = x3 + 2x - 1 has a zero on the interval [0,1]. Homework Problems: pg 133-135: 7, 10, 15-18, 35, 40, 43 Read: Section 2.6 Section 2.6: Limits at Infinity; Horizontal Asymptotes SOLs: APC.1: The student will define and apply the properties of elementary functions, including algebraic, trigonometric, exponential, and composite functions and their inverses, and graph these functions using a graphing calculator. Properties of functions will include domains, ranges, combinations, odd, even, periodicity, symmetry, asymptotes, zeros, upper and lower bounds, and intervals where the function is increasing or decreasing. APC.2: The student will define and apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at infinity, infinite limits, and nonexistent limits. Objectives: Students will be able to: Identify and use limits of functions as x approaches either +/- " Identify horizontal asymptotes of functions Vocabulary: Horizontal Asymptote  a line y = L is a horizontal asymptote, if either limx!" f(x) = L or limx!-" f(x) = L Infinity  " (not a number!! " - " `" 0) Key Concept:  EMBED PowerPoint.Show.8  1. Evaluate: a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3   A horizontal asymptote for a function f is a line y = L such that , , , or both. A function may have at most 2 horizontal asymptotes.  2. Find the horizontal asymptote(s) for a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Homework Problems: pg 146 - 149: 2, 3, 7, 11, 13, 18, 27, 29, 33, 38, 39 Read: Section 2.7 Section 2.7: Tangents, Velocities, and Other Rates of Change SOLs: APC.2: The student will define and apply the properties of limits of functions. This will include limits of a constant, sum, product, quotient, one-sided limits, limits at infinity, infinite limits, and nonexistent limits. APC.12: The student will apply the derivative to solve problems, including tangent and normal lines to a curve, curve sketching, velocity, acceleration, related rates of change, Newton's method, differentials and linear approximations, and optimization problems. Objectives: Students will be able to: Identify the average and instantaneous rates of change Vocabulary: Average rate of change  "y/"x (the slope of the secant line between two points on the curve) Instantaneous rate of change  lim "y/"x ("x!0) (the slope of the tangent line at a point on the curve) Key Concept:  EMBED PowerPoint.Show.8  (Euclid) Given a secant line of a curve, as Q ! P , the secant line approaches the tangent line at P, i.e., the tangent line is the limiting position. If P has coordinates (a, f(a)) and Q has coordinates (a + h, f(a+h)), then  EMBED Equation.3  and  EMBED Equation.3  Example 1: Find the slope of the tangent line to the curve f(x) = 3 - 2x 2x2 where x = 1. Example 2: Find the slope of the tangent line (using the definition) where  EMBED Equation.3  and a = 2.  The expression is the average rate of change of f(x) on the closed interval [a, x]. Thus,  EMBED Equation.3  is the instantaneous rate of change of y = f(x) at x = a. In particular, if y = f(x) is the position of an object at time x,  EMBED Equation.3  is the average velocity and  EMBED Equation.3  is the instantaneous velocity at x = a. Examples: 1. Find the instantaneous velocity at time t = 3 seconds if the particles position at time  EMBED Equation.3  is given by f(t) = t2 + 2t ft. 2. For a particle whose position at time t is f(t) = 6t2 - 4t +1 ft.: a. Find the average velocity over the following intervals: i. [1, 4] ii. [1, 2] iii. [1, 1.2] iv. [1, 1.01] b. Find the instantaneous velocity of the particle at t = 1 sec. Homework Problems: pg 155 - 157: 2, 7, 15, 22, 27 Read: Section 2.8 Chapter 2.8: Derivatives SOLs: APC.4: The student will find the derivative of an algebraic function by using the definition of a derivative. This will include investigating and describing the relationship between differentiability and continuity. Objectives: Students will be able to: Understand the derivative as the slope of the tangent and as a rate of change Vocabulary: None new Key Concept:  The derivative of a function f(x) is another function f(x) whose value at a number a is if the limit exists. Example: Use the definition to find a. f(1) if  EMBED Equation.3  b. f(x) if f(x) = 2x3 x2 + 3x - 1 c. f(x) if  EMBED Equation.3  An alternate form of the derivative is  EMBED Equation.3  Example: Using the alternate form, find f(3) for f(x) = x2 + 10x If f(a) exists, the tangent line to the graph of f(x) at a is the line through (a, f(a)) with slope f(a). If f(a) does not exist, then the tangent line might not exist, might be a vertical tangent line or might not be unique. Left is a graph of the function f(x), place the following quantities in order from lowest to highest. ____ f(a) ____ f(b) ____ slope of the secant line PQ ____ slope of the secant line QR ____ (f(c) f(a)) / (c a) The slope of the tangent line (measured by f(a)) is the same as the instantaneous rate of change of y = f(x) with respect to x at x = a. The Definition of Derivative Reversed: Each of the following is a derivative, but of what function and at what point? Function Point a.  EMBED Equation.3  b.  EMBED Equation.3  c. EMBED Equation.3  d.  EMBED Equation.3  e.  EMBED Equation.3  f.  EMBED Equation.3  g.  EMBED Equation.3  Homework Problems: pg 163 - 164: 4, 13, 19, 29 Read: Section 2.9 Section 2.9: The Derivative as a Function SOLs: APC.4: The student will find the derivative of an algebraic function by using the definition of a derivative. This will include investigating and describing the relationship between differentiability and continuity. Objectives: Students will be able to: Understand the difference between differentiability and continuity Vocabulary: Differentiable a function is differentiable at a point, a, if f(a) exists Key Concept:  EMBED PowerPoint.Show.8  To say that a function f is differentiable at x means that f(x) exists. f is differentiable on an interval if it is differentiable at every number in the interval. Different Notations: f(x) y (not desirabley(x) is good)   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Using the definition of derivative, prove  EMBED Equation.3  is not differentiable at x = 0.  Theorem: If f is differentiable at a point c, then f is continuous at c. The converse is false. * differentiability EMBED Equation.3 continuity  EMBED Equation.3 limit There are 3 common ways for a function to fail to be differentiable at a point (look at page 170): 1. The graph has a sharp point or cusp. Example:  EMBED Equation.DSMT4  2. The function is discontinuous.  Example:  EMBED Equation.DSMT4   3. The graph has a vertical tangent line. Example:  EMBED Equation.DSMT4  Example 1: For the function f(x) pictured below, tell whether the statement is true or false. f(x) is continuous at 0. f(x) is differentiable at 0. f(x) is continuous at 2. f(x) is differentiable at 2. f(x) is continuous at 3. f(x) is differentiable at 3. f(x) is continuous at 4. f(x) is differentiable at 4. Example 2: Given f, draw f .Example 3: Given f, draw f.  EMBED PowerPoint.Slide.8  EMBED PowerPoint.Slide.8  Example 4: Given f, draw f. Example 5: Given f, draw f.  EMBED PowerPoint.Slide.8  EMBED PowerPoint.Slide.8  Symmetric Difference Quotient:  This is the formula programmed into the TIs. The command is nDeriv. The value for h is .01 Examples: Use nDeriv to find the derivative of f(x) = x2 + 1 at x = -1. The derivative of f(x) = x3 is 3x2 so f(2) = 12. Check the result on the calculator. It is important to recognize exact values and approximate values. It is most helpful to use nDeriv with functions that are difficult to differentiate by hand.  Example: Let . Find an equation of the tangent line to the curve at x = -1 using nDeriv to find the slope. We can also graph the derivative of our function. Sometimes the calculator can give incorrect information. Try nDeriv(|x|, x, 0). It might help to look at the graph of the derivative of |x|. Note the derivative of |x| does not exist at 0 because  EMBED Equation.3 . Note that the derivative of |x| is  EMBED Equation.3  Summary: Polynomials are differentiable everywhere. Rational functions and trig functions are differentiable over their domains. Sums, differences, products, powers, quotients, and composites are differentiable where defined. Homework Problems: pg 173 - 175: 4, 5, 12, 19, 21, 24, 37 Read: Review Chapter 2 Chapter 2: Review SOLs: None New Objectives: Students will be able to: Know material presented in Chapter 2 Vocabulary: None new Key Concept: Limits Continuity Intermediate Value Theorem Differentiability Non-Calulator Multiple Choice 1.  EMBED Equation.DSMT4  A. 0 B. 1 C.  EMBED Equation.DSMT4  D. 1 E. does not exist 2. 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Given that  EMBED Equation.DSMT4  find  such that |(2x + 1)  3| d" 0.01 whenever 0 < |x  1| d" . A. 3 B. 0.05 C. 0.005 D. 0.03 E. 0.02 4. Find  EMBED Equation.DSMT4  A. 8 B.  1 C. 1 D.  8 E. does not exist 5. Find  EMBED Equation.DSMT4  A. " B. -" C. 0 D. E. does not exist 6. Find the value of the limit  EMBED Equation.DSMT4  A. 8 B. 4 C. 2 D. 12 E. 6 7. At what value of  EMBED Equation.DSMT4  does the function  EMBED Equation.DSMT4  have a removable discontinuity? A. 3 B. 3 C. 2 D. 1 E. 1 Calculator Multiple Choice 8. Suppose  EMBED Equation.DSMT4  is a function defined on the interval [0,5] such that  EMBED Equation.DSMT4 find  EMBED Equation.DSMT4  A. 8 B.  EMBED Equation.DSMT4  C. 1 D.  EMBED Equation.DSMT4  E. does not exist 9. If  EMBED Equation.DSMT4  and f(x) is continuous on [a, b], then A. f(x) must be identically zero B. f(x) may be different from zero for all x on (a, b) There exists at least one number c, a < c < b, such that f(c) = 0 f(x) must exist for every x on (a, b) 10. The displacement in meters of a particle moving in a straight line is given by s = t + t where t is measured in seconds. Find the average velocity in meters per second over the time period [1, 2]. A. 5 B. 3 C. 8 D. 1 E. 4 11. For the function f(x) whose graph is shown at the right, which statement is false?  A.  EMBED Equation.DSMT4  B.  EMBED Equation.DSMT4  C.  EMBED Equation.DSMT4  D.  EMBED Equation.DSMT4  E.  EMBED Equation.DSMT4  Free Response (1976 AB 2) Given the two functions f and h such that f(x) = x - 3x - 4x + 12 and  EMBED Equation.DSMT4  Find all zeros of the function  EMBED Equation.DSMT4 . 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Tph2MKKKAJAhILX------ % ( Rp@"Arial P@|ϓ Q|Hm| 7ϓ ϓ  !w PD|p|m|z08Hjdv !w PD|p|m|Z|m|dvD !wPD|p|m|Z|m|dv%    TX2MKKKAJAILP=  % ( Rp@"Arial P@|ϓ Q|Hm| 8ϓ ϓ  !w PD|p|m|z08jdv !w PD|p|m|Z|m|dvD !wPD|p|m|Z|m|dv%    T`2MKKKAJAILT---  % ( Rp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    TT.G=bKKKAJA.^LP" % ( Rp@"Arial{L|ϓ Q|Hm|% yϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv !w\P|p|m|Z|m|dv%    T|>GbKKKAJA>^L\x 9  % ( Rp@"Arial@{8L|ϓ Q|Hm|% zϓ ϓ m50l (@{z08hdv !w \P|p|m|Z|m|dv !w\P|p|m|Z|m|dv%    TTGbKKKAJA^LP- % ( Rp@"Arial L|ϓ Q|Hm|% {ϓ ϓ  !w \P|p|m|z080idv !w \P|p|m|Z|m|dv !w\P|p|m|Z|m|dv%    TxGbKKKAJA^L\3 6 % ( F(GDIC FGDICRp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    33TT0?KKKAJA0LP" % ( Rp@"Arial{L|ϓ Q|Hm| Nϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv !w\P|p|m|Z|m|dv%    33T|@KKKAJA@L\y 5  % ( Rp@"Arial@{8L|ϓ Q|Hm| Oϓ ϓ m50l (@{z08hdv !w \P|p|m|Z|m|dv !w\P|p|m|Z|m|dv%    33TTKKKAJALP- % ( Rp@"Arial L|ϓ Q|Hm| Pϓ ϓ  !w \P|p|m|z080idv !w \P|p|m|Z|m|dv !w\P|p|m|Z|m|dv%    33T|KKKAJAL\2 3 % ( Rp@"Arial0 `<f`k>0#0-i *`k0-z08gT0I0i0U-i @{dvdvD0lx8v D\t(0dv%    33TTKKKAJALPm % ( Rp@"Ariali *`k@|ϓ Q|Hm|% ϓ ϓ I0i0U-i @{z08g0l D\t(0dvD !wPD|p|m|Z|m|dv%    33TTKKKAJALP3  % ( Rp@"Ariali0U-@|ϓ Q|Hm|% ϓ ϓ 0l D\t(z08hdv !w PD|p|m|Z|m|dvD !wPD|p|m|Z|m|dv%    33TX,KKKAJALP=  % ( Rp@"Arial @|ϓ Q|Hm|% ϓ ϓ  !w PD|p|m|z08hdv !w PD|p|m|Z|m|dvD !wPD|p|m|Z|m|dv%    33Td-LKKKAJA-LT---- % ( Rp@"Arial P@|ϓ Q|Hm|% ϓ ϓ  !w PD|p|m|z080idv !w PD|p|m|Z|m|dvD !wPD|p|m|Z|m|dv%    33TXTiKKKAJATLP=  % ( Rp@"Arial P@|ϓ Q|Hm|% ϓ ϓ  !w PD|p|m|z08idv !w PD|p|m|Z|m|dvD !wPD|p|m|Z|m|dv%    33TpjKKKAJAjLX------ % ( Rp@"Arial P@|ϓ Q|Hm|% ϓ ϓ  !w PD|p|m|z08Hjdv !w PD|p|m|Z|m|dvD !wPD|p|m|Z|m|dv%    33T`KKKAJALT=  % ( Rp@"Arial P@|ϓ Q|Hm|% ϓ ϓ  !w PD|p|m|z08jdv !w PD|p|m|Z|m|dvD !wPD|p|m|Z|m|dv%    33T`KKKAJALT---  % ( Rp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    33TT0?KKKAJA0LP" % ( Rp@"Arial{L|ϓ Q|Hm| kϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv !w\P|p|m|Z|m|dv%    33T|@KKKAJA@L\x 6  % ( Rp@"Arial@{8L|ϓ Q|Hm| lϓ ϓ m50l 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PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TdKKKAJALT----  % ( Rp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    TT/>KKKAJA/LP" % ( Rp@"Arial{L|ϓ Q|Hm|w ոϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv!\P|p|m|Z|m|dv%    Tx?KKKAJA?L\x 12  % ( Rp@"Arial@{8L|ϓ Q|Hm|w ָϓ ϓ m50l (@{z08hdv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TTKKKAJALP- % ( Rp@"Arial L|ϓ Q|Hm|w ׸ϓ ϓ ! \P|p|m|z080idv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    T|KKKAJAL\0 12 % ( F(GDIChFGDICRp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    TT.=8KKKAJA.4LP" % ( Rp@"Arial{L|ϓ Q|Hm| ϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv!\P|p|m|Z|m|dv%    Tx>8KKKAJA>4L\y 10  % ( Rp@"Arial@{8L|ϓ Q|Hm| ϓ ϓ m50l (@{z08hdv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TT8KKKAJA4LP- % ( Rp@"Arial L|ϓ Q|Hm| ϓ ϓ ! \P|p|m|z080idv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    Tx8KKKAJA4L\1 1 % ( Rp@"Arial0 <f`k>0#0-&i *`k0-z08gT0I0i0U-&i @{dvdvD0lx8v D\t(0dv%    TT2MKKKAJAILPm % ( Rp@"Ariali *`k@|ϓ Q|Hm|w ϓ ϓ I0i0U-&i @{z08g0l D\t(0dvD!PD|p|m|Z|m|dv%    TTA SKKKAJAPLP2  % ( Rp@"Ariali0U-@|ϓ Q|Hm|w ϓ ϓ 0l D\t(z08hdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TX2*MKKKAJAILP=  % ( Rp@"Arial @|ϓ Q|Hm|w ϓ ϓ ! PD|p|m|z08hdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    Td+2JMKKKAJA+ILT---- % ( Rp@"Arial P@|ϓ Q|Hm|w ϓ ϓ ! PD|p|m|z080idv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TXR2gMKKKAJARILP=  % ( Rp@"Arial P@|ϓ Q|Hm|w ϓ ϓ ! PD|p|m|z08idv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    Tph2MKKKAJAhILX------ % ( Rp@"Arial P@|ϓ Q|Hm|w ϓ ϓ ! PD|p|m|z08Hjdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TX2MKKKAJAILP=  % ( Rp@"Arial P@|ϓ Q|Hm|w ϓ ϓ ! PD|p|m|z08jdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    T`2MKKKAJAILT---  % ( Rp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    TT.G=bKKKAJA.^LP" % ( Rp@"Arial{L|ϓ Q|Hm| ϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv!\P|p|m|Z|m|dv%    T|>GbKKKAJA>^L\x 9  % ( Rp@"Arial@{8L|ϓ Q|Hm| ϓ ϓ m50l (@{z08hdv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TTGbKKKAJA^LP- % ( Rp@"Arial L|ϓ Q|Hm| ϓ ϓ ! \P|p|m|z080idv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TxGbKKKAJA^L\0 1 % ( F(GDIC FGDICRp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    33TT0?KKKAJA0LP" % ( Rp@"Arial{L|ϓ Q|Hm|w ϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv!\P|p|m|Z|m|dv%    33T|@KKKAJA@L\y 5  % ( Rp@"Arial@{8L|ϓ Q|Hm|w ϓ ϓ m50l (@{z08hdv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33TTKKKAJALP- % ( Rp@"Arial L|ϓ Q|Hm|w ϓ ϓ ! \P|p|m|z080idv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33T|KKKAJAL\1 4 % ( Rp@"Arial0w <f`k>0#0-Hi *`k0-z08gT0I0i0U-Hi @{dvdvD0lx8v D\t(0dv%    33TTKKKAJALPm % ( Rp@"Ariali *`k@|ϓ Q|Hm| :ϓ ϓ I0i0U-Hi @{z08g0l D\t(0dvD!PD|p|m|Z|m|dv%    33TTKKKAJALP3  % ( Rp@"Ariali0U-@|ϓ Q|Hm| ;ϓ ϓ 0l D\t(z08hdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    33TX,KKKAJALP=  % ( Rp@"Arial @|ϓ Q|Hm| <ϓ ϓ ! PD|p|m|z08hdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    33Td-LKKKAJA-LT---- % ( Rp@"Arial P@|ϓ Q|Hm| =ϓ ϓ ! PD|p|m|z080idv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    33TXTiKKKAJATLP=  % ( Rp@"Arial P@|ϓ Q|Hm| >ϓ ϓ ! PD|p|m|z08idv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    33TpjKKKAJAjLX------ % ( Rp@"Arial P@|ϓ Q|Hm| ?ϓ ϓ ! PD|p|m|z08Hjdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    33T`KKKAJALT=  % ( Rp@"Arial P@|ϓ Q|Hm| @ϓ ϓ ! PD|p|m|z08jdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    33T`KKKAJALT---  % ( Rp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    33TT0?KKKAJA0LP" % ( Rp@"Arial{L|ϓ Q|Hm|w +ϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv!\P|p|m|Z|m|dv%    33T|@KKKAJA@L\x 6  % ( Rp@"Arial@{8L|ϓ Q|Hm|w ,ϓ ϓ m50l (@{z08hdv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33TTKKKAJALP- % ( Rp@"Arial L|ϓ Q|Hm|w -ϓ ϓ ! \P|p|m|z080idv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33T|KKKAJAL\0 6 % ( F(GDIC)BoFGDICRp@"Arial04\m04{m> 0z08g8t@{8 0@{0$l`p`0lTq(@{(dv%    T4IiKKKAJA4cLTangent versus Secant Lines    % ( F(GDICMTFEMF+*@$??@, D1#CD1#C{DIR$C{D9%C{D('CD:(CD:(CuD:(CǽD('CǽD9%CǽDIR$CuD1#CD1#C@,( !b $$=='%  ;UP %, $, $> $T $j $| %| %| (%j (%T (%> %, %, <>MS % $$AAF`TEMF+@-@4^UU?A@,-$$==_888% % UPKT %, $, $> $T $j $| %| %| (%j (%T (%> %, %, % % $$AA( " FEMF+@ FGDICF(GDIC|FGDICRp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    TT\|kKKKAJA\LP" % ( Rp@"Arial{L|ϓ Q|Hm|w =ϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv!\P|p|m|Z|m|dv%    Txl|KKKAJAlL\y 5  % ( Rp@"Arial@{8L|ϓ Q|Hm|w >ϓ ϓ m50l (@{z08hdv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TT|KKKAJALP- % ( Rp@"Arial L|ϓ Q|Hm|w ?ϓ ϓ ! \P|p|m|z080idv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    T||KKKAJAL\1 4 % ( Rp@"Arial0w O<f`k>0#0-i *`k0-z08gT0I0i0U-i @{dvdvD0lx8v D\t(0dv%    TT2KKKAJALPm % ( Rp@"Ariali *`k@|ϓ Q|Hm| kϓ ϓ I0i0U-i @{z08g0l D\t(0dvD!PD|p|m|Z|m|dv%    TT36KKKAJA3LPt % ( Rp@"Ariali0U-@|ϓ Q|Hm| lϓ ϓ 0l D\t(z08hdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TX>SKKKAJA>LP=  % ( Rp@"Arial @|ϓ Q|Hm| mϓ ϓ ! PD|p|m|z08hdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TdTsKKKAJATLT---- % ( Rp@"Arial P@|ϓ Q|Hm| nϓ ϓ ! PD|p|m|z080idv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TX{KKKAJA{LP=  % ( Rp@"Arial P@|ϓ Q|Hm| oϓ ϓ ! PD|p|m|z08idv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TpKKKAJALX------  % ( Rp@"Arial P@|ϓ Q|Hm| pϓ ϓ ! PD|p|m|z08Hjdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    TXKKKAJALP=  % ( Rp@"Arial P@|ϓ Q|Hm| qϓ ϓ ! PD|p|m|z08jdv! PD|p|m|Z|m|dvD!PD|p|m|Z|m|dv%    T`KKKAJALT--- % ( Rp@"Arial0.\m04{m> 0z08g8t@{8 0@{0$l`pm50l s(@{xdv%    TT\kKKKAJA\LP" % ( Rp@"Arial{L|ϓ Q|Hm|w Zϓ ϓ 8t@{8 0@{0z08hm50l (@{xdv!\P|p|m|Z|m|dv%    TxlKKKAJAlL\x 6  % ( Rp@"Arial@{8L|ϓ Q|Hm|w [ϓ ϓ m50l (@{z08hdv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TTKKKAJALP- % ( Rp@"Arial L|ϓ Q|Hm|w \ϓ ϓ ! \P|p|m|z080idv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    T|KKKAJAL\0 6 % ( F(GDIC.FtEMF+*@$??@.<0CE6CgCȖCeD?[CȵDHC@.( !b $$=='%  % V,.E! & %  % $$AA" FEMF+@ FGDICRp%%@"Arial0#\m04{m> 0z08g8t@{8 0@{0$l`pm50lq(@{(dv%    T|&KKKAJA L\Secant 1   % ( F4(EMF+*@$??!b Ld)??" 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M}@p줐z^`r:aXKs5V=;&[nXy+NeYX 5 97ږX,ks30 h]:>+=HLVJM^4u>+rvtmfh?qP 6aAg|UP\=h=3u+uW@njVڶ% qϪUiiq%SC_=G/5'^h,!s]OEiovN7.32/ yhxNJsc[}^y͸jWXFm/(jhJos_5ɂNm;4PuƋN1䙵qWmlzITmL񢍶6rg&\/Q<9fj{ylU3khfv΋'}v/Odc^_Im ylB dsf愗>9Ϧ9>^t >^}ylȨ&$ R bL>_MjpUǼb >4}6ų[neOW>s,sh/KHcnԘT$u>+rCJ^vv']1:/UnvNeU袲gMrSghvnܔi9H;{Π,Ɣ!7tq3Ƿst1e^t1<1%mn i;R@O|KRa^.܆]q:y-vжN%K)UAlNh[,KRxQK@P)y)N9>/ejjP!Uh:Rt1 CY]r|^Լ66&{FU9=/45,U3khfV΋'r|^j~/Xj#0Km,M^rR^t}5վw47匜25$/¨V-/'@U?üg`ː0&sUsUF5_-'^5^9?sRX9\9vNd0'DZ<tOOy*жc/A;ZHos2夸u9 .{E vN4R^sR>[+CT!Y+/ɘ)sz;GWF8E#W+%3lkn hKrSj<ܔ+waw#v:mM9f̔,x9sS=\#.ж߳K1fJޜ]񢰗x]71S6PnJ:^_1246863337dO)&U-UOle &EPRINT=xCompObj'{  l1'% EMFx\KhCF, EMF+@F\PEMF+"@ @ $@ 0@?!@ @ F(GDICFXLEMF+*@$?? @$yD;D!b $$==% % V0.x>.x>% % $$AA" FEMF+@ FGDICF(GDIC1FGDICF(GDICUFGDICF(GDIC&FFGDICRp@"Arial04\4 0D$D4> 0z08|8tP8 0P0$ܘĨm50, Ĩ(P( Ĩdv%    Tp @KKKAJA:LXLimits  % ( F(GDICIFGDICRp@"Arial0u o<8>0#0-\08TKY0-z08|T0I0i0T-\PĨdvdvT@"Arialwz Hz dv%    Tl!P&iKKKAJA!e0LWhen we look at the limit below, we examine the             % ( Rp@"Arial\08@| Q|Hm|z I0i0T-\Pz08T @"Arialwz z dv"!PD|p|m|Z|m|dv%    T`'PAiKKKAJA'eLTf(x  % ( Rp@"Ariali0T-@| Q|Hm|z T @"Ariaz08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    TBPiKKKAJABeLt) values as x gets        % ( Rp@"Arial @| Q|Hm|z "! PD|p|m|z08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    TdP-iKKKAJAeLTvery  % ( &% h6.h% ( Rp@"Arial P@| Q|Hm|z d|0Q|Hm|z0m|z08dv"! PD|p|m|Z| m|dv"!PD|p|m|Ztm|dv%    T5PiKKKAJA5e Ldclose to a:     % ( Rp@"Arial0t  0 #DP5TiK0W0 ~D#DPz08| 60P!0ܘĨm50䚥Ĩ(PdPĨdv%    T!{KKKAJA!Ltread: the limit of    % ( Rp@"ArialW0 L|Q|Hm|t 0P!0z08Ĩm50 Ĩ(PdPdv"!\P|p|m|Z|m|dv%    T`{KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|t m50 Ĩ(Pdz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    T{/KKKAJAL), as x approaches a, equals L           % ( Rp@"Arial0u <>0#0-\00-z08|T0I0i0T-\PĨdvdvT@"Arialwt Ht dv%    T`!KKKKAJA!LTOne  % ( Rp@"Arial\0@| Q|Hm|t ̐I0i0T-\Pz08T @"Arialwt t dv"!PD|p|m|Z|m|dv%    TTLRKKKAJALLP- % ( Rp@"Ariali0T-@| Q|Hm|t ͐T @"Ariaz08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    TSKKKAJAS LdSided Limits   % ( &% !6% ( Rp@"Arial @| Q|Hm|t d|0Q|Hm|t0ϐm|z08 dv"! PD|p|m|Z| m|dv"!PD|p|m|Ztm|dv%    TTKKKAJALP: % ( Rp@"Arial0u 80 $DP@TiK0W08D$DPz08|4A0P!0ܘĨm50Ĩ(PxĨdv%    Td!HKKKAJA!LTLeft  % ( Rp@"ArialW08L|Q|Hm|u 0P!0z08Ĩm50 Ĩ(Pxdv"!\P|p|m|Z|m|dv%    TTIPKKKAJAILP-33 % ( Rp@"ArialL|Q|Hm|u m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    TtQGKKKAJAQ1Lhand limit (as x approaches a from the left side             % ( Rp@"Arial ĨL|Q|Hm|u "! \P|p|m|z08 dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    TTHSKKKAJAHLP   % ( Rp@"Arial \L|Q|Hm|u "! \P|p|m|z08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    T|[KKKAJA[L\smaller)    % ( Rp@"Arial0u 80 $DPATiK0W08D$DPz08|4B0P!0ܘĨm50Ĩ(PxĨdv%    Tl!XKKKAJA!LXRIght  % ( Rp@"ArialW08L|Q|Hm|u 0P!0z08Ĩm50 Ĩ(Pxdv"!\P|p|m|Z|m|dv%    TTY`KKKAJAYLP- % ( Rp@"ArialL|Q|Hm|u m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    TxagKKKAJAa2Lhand limit (as x approaches a from the right side              % ( Rp@"Arial ĨL|Q|Hm|u "! \P|p|m|z08 dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    TThsKKKAJAhLP  % ( Rp@"Arial \L|Q|Hm|u "! \P|p|m|z08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    Tx{KKKAJA{L\larger)   % ( Rp@"Arial0u 80 \$DPRTiK0W08D\$DPz08|4S0P!0ܘĨm50Ĩ(PxĨdv%    Tx!6tOKKKAJA!KL\The two   % ( Rp@"ArialW08L|Q|Hm|u 0P!0z08Ĩm50 Ĩ(Pxdv"!\P|p|m|Z|m|dv%    TTu6{OKKKAJAuKLP- % ( Rp@"ArialL|Q|Hm|u m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    T|6OKKKAJA|K9Lsided limit (first one shown) = L if and only if both one                % ( Rp@"Arial ĨL|Q|Hm|u "! \P|p|m|z08 dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    TT6OKKKAJAKLP- % ( Rp@"Arial \L|Q|Hm|u "! \P|p|m|z08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    T6`OKKKAJAKLlsided limits = L      % ( Rp@"Arial0u 0 \$DPRTiK0W0D\$DPz08|S0P!0ܘĨm50Ĩ(P<<Ĩdv%    Tb{KKKAJAwLif and only if     % ( Rp@"ArialW0L|Q|Hm|u 0P!0z08Ĩm50 Ĩ(P<<dv"!\P|p|m|Z|m|dv%    T`:b`{KKKAJA:wLTand   % ( Rp@"Arial0z <>0#0-\0d\0-z08|T0I0i0T-\PĨdvdvT@"Arialwu Hu dv%    T!KKKAJA!LtVertical Asymptotes       % ( &% !6% ( Rp@"Arial\0@| Q|Hm|u d|0Q|Hm|u0Pz08T @"Arialw u dv"!PD|p|m|Ztm|dv%    T` KKKAJALT:  % ( Rp@"Arial|/0\^P4^\/0]z08|4044]ܘĨm50@fĨ(PdfĨdv%    T!"KKKAJA!LThe line x = a is called a           % ( Rp@"Arial^L|Q|Hm|z ֐044]z08Ĩm50 fĨ(Pdfdv"!\P|p|m|Z|m|dv%    T|#qKKKAJA#L\verticle    % ( Rp@"Arial044L|Q|Hm|z אm50 fĨ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    Tx.KKKAJAxLpasymptote of y =       % ( Rp@"Arial fĨL|Q|Hm|z ؐ"! \P|p|m|z08 dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    T`/IKKKAJA/LTf(x  % ( Rp@"Arial \L|Q|Hm|z ِ"! \P|p|m|z08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    T0JKKKAJAJ&L) if at least one of the following is         % ( Rp@"Arial0t 0 (#DPTiK0W0D(#DPz08|䜥0P!0ܘĨm50,Ĩ(P(Ĩdv%    Tl!PKKKAJA!LXtrue:  % ( F(GDICokFGDICRp@"Arial0z 80 H#DP TiK0W08.DH#DPz0840P!0ܘĨm50 Ĩ(Px Ĩdv%    33T`uKKKAJALTlim % ( Rp@"ArialW08L|Q|Hm|z 0P!0z08 Ĩm50 Ĩ(Px dv"!\P|p|m|Z|m|dv%    33T`uKKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Tlu9KKKAJALX) = L % ( Rp@"Arial0t 0 @#DP TiK0W0*D@#DPz08 0P!0ܘĨm50@Ĩ(PdĨdv%    33TTKKKAJALPx % ( Rp@WingdingsW0L|Q|Hm|t 0P!0z08Ĩm50 Ĩ(Pddv"!\P|p|m|Z|m|dv%    33TTKKKAJALP % ( Rp@"ArialL|Q|Hm|t m50 Ĩ(Pz08 dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TTKKKAJALPa % ( F(GDICFGDICRp@"Arial0z 80 H#DP TiK0W08.DH#DPz0840P!0ܘĨm50 Ĩ(Px Ĩdv%    33T`KKKAJALTlim % ( Rp@"ArialW08L|Q|Hm|z 0P!0z08 Ĩm50 Ĩ(Px dv"!\P|p|m|Z|m|dv%    33T`6KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Tl7jKKKAJA7LX) = L % ( Rp@"Arial0z <8>0#0- \080-z08T0I0i0T- \PĨdvdvT@"Arialwt Ht dv%    33TT KKKAJALPx % ( Rp@Wingdings\08@| Q|Hm|t I0i0T- \Pz08T @"Arialwt t dv"!PD|p|m|Z|m|dv%    33TT KKKAJA LP % ( Rp@"Ariali0T-@| Q|Hm|t T @"Ariaz08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT"KKKAJALPa % ( Rp@"Arial @| Q|Hm|t !"! PD|p|m|z08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT#%KKKAJA#LP- % ( F(GDIC4FGDICRp@"Arial0z 80 H#DP TiK0W08.DH#DPz0840P!0ܘĨm50 Ĩ(Px Ĩdv%    33T`KKKAJALTlim % ( Rp@"ArialW08L|Q|Hm|z 0P!0z08 Ĩm50 Ĩ(Px dv"!\P|p|m|Z|m|dv%    33T`7KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Tl8kKKKAJA8LX) = L % ( Rp@"Arial0z &<8>0#0- \080-z08T0I0i0T- \PĨdvdvT@"Arialwt 8Ht 8dv%    33TT-KKKAJA+LPx % ( Rp@Wingdings\08@| Q|Hm|t 9I0i0T- \Pz08T @"Arialwt 8t 8dv"!PD|p|m|Z|m|dv%    33TT-KKKAJA+LP % ( Rp@"Ariali0T-@| Q|Hm|t :T @"Ariaz08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT#-KKKAJA+LPa % ( Rp@"Arial @| Q|Hm|t ;"! PD|p|m|z08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT$)(KKKAJA$'LP+ % ( F(GDIC?UFGDICRp@"Arial0z 180 H#DP TiK0W08.DH#DPz0840P!0ܘĨm50 Ĩ(Px Ĩdv%    33T`Q[sxKKKAJAQsLTlim % ( Rp@"ArialW08L|Q|Hm|z 20P!0z08 Ĩm50 Ĩ(Px dv"!\P|p|m|Z|m|dv%    33T`{[xKKKAJA{sLTf(x  % ( Rp@"ArialL|Q|Hm|z 3m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Tl[xKKKAJAsLX) = L % ( Rp@"Arial0t P0 @#DP TiK0W0*D@#DPz08 0P!0ܘĨm50@Ĩ(PdĨdv%    33TTizpKKKAJAiLPx % ( Rp@WingdingsW0L|Q|Hm|t Q0P!0z08Ĩm50 Ĩ(Pddv"!\P|p|m|Z|m|dv%    33TTp{}KKKAJApLP % ( Rp@"ArialL|Q|Hm|t Rm50 Ĩ(Pz08 dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TT~zKKKAJA~LPa % ( F(GDICUTFGDICRp@"Arial0z H80 H#DP TiK0W08.DH#DPz0840P!0ܘĨm50 Ĩ(Px Ĩdv%    33T`[xKKKAJAsLTlim % ( Rp@"ArialW08L|Q|Hm|z I0P!0z08 Ĩm50 Ĩ(Px dv"!\P|p|m|Z|m|dv%    33T`[xKKKAJAsLTf(x  % ( Rp@"ArialL|Q|Hm|z Jm50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Tl["xKKKAJAsLX) = L % ( Rp@"Arial0z V<8>0#0- \080-z08T0I0i0T- \PĨdvdvT@"Arialwt iHt idv%    33TTzKKKAJALPx % ( Rp@Wingdings\08@| Q|Hm|t jI0i0T- \Pz08T @"Arialwt it idv"!PD|p|m|Z|m|dv%    33TT{KKKAJALP % ( Rp@"Ariali0T-@| Q|Hm|t kT @"Ariaz08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TTzKKKAJALPa % ( Rp@"Arial @| Q|Hm|t l"! PD|p|m|z08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TTyKKKAJALP- % ( F(GDIC`U FGDICRp@"Arial0z a80 H#DP TiK0W08.DH#DPz0840P!0ܘĨm50 Ĩ(Px Ĩdv%    33T`r[xKKKAJArsLTlim % ( Rp@"ArialW08L|Q|Hm|z b0P!0z08 Ĩm50 Ĩ(Px dv"!\P|p|m|Z|m|dv%    33T`[xKKKAJAsLTf(x  % ( Rp@"ArialL|Q|Hm|z cm50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Tl[xKKKAJAsLX) = L % ( Rp@"Arial0z o<8>0#0- \080-z08T0I0i0T- \PĨdvdvT@"Arialwt Ht dv%    33TTzKKKAJALPx % ( Rp@Wingdings\08@| Q|Hm|t I0i0T- \Pz08T @"Arialwt t dv"!PD|p|m|Z|m|dv%    33TT{KKKAJALP % ( Rp@"Ariali0T-@| Q|Hm|t T @"Ariaz08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TTzKKKAJALPa % ( Rp@"Arial @| Q|Hm|t "! PD|p|m|z08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TTyKKKAJALP+ % ( F(GDIC^FGDICRp@"Arial0z |Ē0 H#DP TiK0W0Ē.DH#DPz08H0P!0ܘĨm50Ĩ(PܩĨdv%    33T`pKKKAJApLTlim % ( Rp@"ArialW0ĒL|Q|Hm|z }0P!0z08 Ĩm50 Ĩ(Pܩdv"!\P|p|m|Z|m|dv%    33T`KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z ~m50 Ĩ(Pܩz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TdKKKAJALT) =  % ( Rp@"Arial ĨL|Q|Hm|z "! \P|p|m|z088dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TTKKKAJALP" % ( Rp@"Arial0t 0 @#DP TiK0W0*D@#DPz08 0P!0ܘĨm50@Ĩ(PdĨdv%    33TTKKKAJALPx % ( Rp@WingdingsW0L|Q|Hm|t 0P!0z08Ĩm50 Ĩ(Pddv"!\P|p|m|Z|m|dv%    33TTKKKAJALP % ( Rp@"ArialL|Q|Hm|t m50 Ĩ(Pz08 dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TTKKKAJALPa % ( F(GDICsFGDICRp@"Arial0z 80 H#DP TiK0W08.DH#DPz0840P!0ܘĨm50Ĩ(PxĨdv%    33T`KKKAJALTlim % ( Rp@"ArialW08L|Q|Hm|z 0P!0z08 Ĩm50 Ĩ(Pxdv"!\P|p|m|Z|m|dv%    33T` KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33 Td2KKKAJALT) =   TT3DKKKAJALP"  % ( Rp@"Arial0z <8>0#0- \080-z08T0I0i0T- \PĨdvdvT@"Arialwt Ht dv%    33TTKKKAJALPx % ( Rp@Wingdings\08@| Q|Hm|t I0i0T- \Pz08T @"Arialwt t dv"!PD|p|m|Z|m|dv%    33TTKKKAJALP % ( Rp@"Ariali0T-@| Q|Hm|t T @"Ariaz08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TTKKKAJALPa % ( Rp@"Arial @| Q|Hm|t "! PD|p|m|z08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TTKKKAJALP- % ( F(GDICFGDICRp@"Arial0z 80 H#DP TiK0W08.DH#DPz0840P!0ܘĨm50Ĩ(PxĨdv%    33T`KKKAJALTlim % ( Rp@"ArialW08L|Q|Hm|z 0P!0z08 Ĩm50 Ĩ(Pxdv"!\P|p|m|Z|m|dv%    33T`;KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33< Td<`KKKAJA<LT) =   TTarKKKAJA<LP"  % ( Rp@"Arial0z <8>0#0- \080-z08T0I0i0T- \PĨdvdvT@"Arialwt Ht dv%    33TT KKKAJA LPx % ( Rp@Wingdings\08@| Q|Hm|t ѐI0i0T- \Pz08T @"Arialwt t dv"!PD|p|m|Z|m|dv%    33TTKKKAJALP % ( Rp@"Ariali0T-@| Q|Hm|t ҐT @"Ariaz08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT 'KKKAJA LPa % ( Rp@"Arial @| Q|Hm|t Ӑ"! PD|p|m|z08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT(-KKKAJA(LP+ % ( F(GDIC^9FGDICRp@"Arial0z P0 L#DPTiK0W0P0DL#DPz08\0P!0ܘĨm50T Ĩ(P Ĩdv%    33T`p!KKKAJApLTlim % ( Rp@"ArialW0PL|Q|Hm|z ː0P!0z08 Ĩm50 Ĩ(P dv"!\P|p|m|Z|m|dv%    33T`!KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z ̐m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Td!KKKAJALT) =  % ( Rp@"Arial ĨL|Q|Hm|z ͐"! \P|p|m|z088dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TT!KKKAJALP- % ( Rp@"Arial \L|Q|Hm|z ΐ"! \P|p|m|z08Ēdv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TT!KKKAJALP" % ( Rp@"Arial0t 0 @#DP TiK0W0*D@#DPz08 0P!0ܘĨm50@Ĩ(PdĨdv%    33TT#2KKKAJA0LPx % ( Rp@WingdingsW0L|Q|Hm|t 0P!0z08Ĩm50 Ĩ(Pddv"!\P|p|m|Z|m|dv%    33TT$2KKKAJA0LP % ( Rp@"ArialL|Q|Hm|t m50 Ĩ(Pz08 dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TT#2KKKAJA0LPa % ( F(GDICs9FGDICRp@"Arial0z P0 L#DPTiK0W0P0DL#DPz08\0P!0ܘĨm50T Ĩ(P Ĩdv%    33T`!KKKAJALTlim % ( Rp@"ArialW0PL|Q|Hm|z 0P!0z08 Ĩm50 Ĩ(P dv"!\P|p|m|Z|m|dv%    33T` !KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Td2!KKKAJALT) =  % ( Rp@"Arial ĨL|Q|Hm|z "! \P|p|m|z088dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TT3:!KKKAJA3LP- % ( Rp@"Arial \L|Q|Hm|z "! \P|p|m|z08Ēdv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TT;L!KKKAJA;LP" % ( Rp@"Arial0z <8>0#0- \080-z08T0I0i0T- \PĨdvdvT@"Arialwt Ht dv%    33TT#2KKKAJA0LPx % ( Rp@Wingdings\08@| Q|Hm|t I0i0T- \Pz08T @"Arialwt t dv"!PD|p|m|Z|m|dv%    33TT$2KKKAJA0LP % ( Rp@"Ariali0T-@| Q|Hm|t T @"Ariaz08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT#2KKKAJA0LPa % ( Rp@"Arial @| Q|Hm|t "! PD|p|m|z08dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT"-KKKAJA,LP- % ( F(GDIC9FGDICRp@"Arial0z P0 L#DPTiK0W0P0DL#DPz08\0P!0ܘĨm50T Ĩ(P Ĩdv%    33T`!KKKAJALTlim % ( Rp@"ArialW0PL|Q|Hm|z 0P!0z08 Ĩm50 Ĩ(P dv"!\P|p|m|Z|m|dv%    33T`;!KKKAJALTf(x  % ( Rp@"ArialL|Q|Hm|z m50 Ĩ(Pz08dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33Td<`!KKKAJA<LT) =  % ( Rp@"Arial ĨL|Q|Hm|z "! \P|p|m|z088dv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TTah!KKKAJAaLP- % ( Rp@"Arial \L|Q|Hm|z "! \P|p|m|z08Ēdv"! \P|p|m|Z|m|dv"!\P|p|m|Z|m|dv%    33TTiz!KKKAJAiLP" % ( Rp@"Arial0z <8>0#0- \080-z08T0I0i0T- \PĨdvdvT@"Arialwt %Ht %dv%    33TT #2KKKAJA 0LPx % ( Rp@Wingdings\08@| Q|Hm|t &I0i0T- \Pz08T @"Arialwt %t %dv"!PD|p|m|Z|m|dv%    33TT$2KKKAJA0LP % ( Rp@"Ariali0T-@| Q|Hm|t 'T @"Ariaz08 dv"! PD|p|m|Z|m|dv"!PD|p|m|Z|m|dv%    33TT #'2KKKAJA 0LPa % ( Rp@"Arial @| Q|Hm|t ("! PD|p|m|z08dv"! 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S  0 ^ `  >*  0 ^   @*  0l ^ `  @*H  0޽h ? 3380___PPT10.K tv Default Design  (   E <0 ]  8Limits   6t  When we look at the limit below, we examine the f(x) values as x gets very close to a: read: the limit of f(x), as x approaches a, equals L One-Sided Limits: Left-hand limit (as x approaches a from the left side  smaller) RIght-hand limit (as x approaches a from the right side  larger) The two-sided limit (first one shown) = L if and only if both one-sided limits = L if and only if and Vertical Asymptotes: The line x = a is called a verticle asymptote of y = f(x) if at least one of the following is true:IF  7 Thb09t,4  6pХb 2 lim f(x) = L xa:   >J  6ץ[| 4 lim f(x) = L xa-N    >J  64 4 lim f(x) = L xa+N    >4  6hp 2 lim f(x) = L xa:   >J  6 d  4 lim f(x) = L xa-N    >J  6\  4 lim f(x) = L xa+N    >D  6YX $n  2 lim f(x) = " xaJ $   >J  6/YX   4 lim f(x) = " xa-N    >J  6H(YX   4 lim f(x) = " xa+N    >F  6\~ $n  4 lim f(x) = -" xaJ $   >L  64\~   6 lim f(x) = -" xa-N    >L  6!\~   6 lim f(x) = -" xa+N    >H  0޽h ? 3380___PPT10.K tvr0s e% Slide Titles dO)"Microsoft PowerPoint PresentationMSPresentationPowerPoint.Show.89q %_O7 `Chris HeadleeChris HeadleeOh+'0 XEPRINT#%2CompObj:{ObjInfo$(<Pictures                          ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  l1'% EMF2\KhCF, EMF+@F\PEMF+"@ @ $@ 0@?!@ @ F(GDICFXLEMF+*@$?? @$yD;D!b $$=='% % V0.x>.x>% % $$AA" FEMF+@ FGDICF(GDIC1FGDICF(GDICUFGDICF(GDIC&FFGDICRp@"Arial04\8$0ХT8> 0z088tT8 0T0$Ȩm500 Ȩ(T( Ȩdv%    Tp @KKKAJA:LXLimits  % ( F(GDICO+F(GDIC X*F(GDIC FEMF+*@$??@@4^UU?A@, CwD6DwD@!b $$==_888% % W$ !-!% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@, CU}C6DU}C@!b $$==_888% % W$ H-H% % $$AA( " FEMF+@ FGDICF(GDIC WYFEMF+*@$??@@4^UU?A@, C9C6D9C@!b $$==_888% % W$ UYw-w% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@, C {C6D {C@!b $$==_888% % W$ -% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@ , C(C6D(C@ !b $$==_888% % W$  - % % $$AA( " FEMF+@ FGDICF(GDIC XZFEMF+*@$??@ @4^UU?A@ , CUB6DUB@  !b $$==_888% % W$ VZ-% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@ @4^UU?A@ , CC6DC@  !b $$==_888% % W$ .-.% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@, C9C6D9C@!b $$==_888% % W$ `-`% % $$AA( " FEMF+@ FGDICF(GDIC )+FEMF+*@$??@@4^UU?A@, CȔC6DȔC@!b $$==_888% % W$ '+-% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@, CLC6DLC@!b $$==_888% % W$  - % % $$AA( " FEMF+@ FGDICF(GDIC oqFEMF+*@$??@@4^UU?A@, CCB6DCB@!b $$==_888% % W$ mq-% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@, CC6DC@!b $$==_888% % W$ -% % $$AA( " FEMF+@ FGDICF(GDIC npFEMF+*@$??@@4^UU?A@, CC6DC@!b $$==_888% % W$ mq-% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@, C'C6D'C@!b $$==_888% % W$ -% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@, Crl5C6Drl5C@!b $$==_888% % W$ O -O % % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@@4^UU?A@, CrLC6DrLC@!b $$==_888% % W$ -% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@ @4^UU?A@!, CC6DC@! 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p-E`t7 I"@% % $$AA( " FEMF+@ FGDICF(GDICT]FEMF+*@$??@  C9fCNC9fCUC2-CUC"CUC$CNC߭CC߭C@C߭C9~C$C9~C"C9~C2-C@C9fCC9fC@ ( !b $$=='%  ;UP EE^}}^EE<>T\ % $$AAF`TEMF+@!@4^UU?A33@ !$$==_88833% % UPR] EE^}}^EE% % $$AA( " FEMF+@ FGDICF(GDIC UFEMF+*@$??@"4(H@A33@#XLUuDC#DU]CDqCDCU1$DÇC"DHCU#DB@#"!b $$==_888233% % U8T"RA$&<')(}(p% % $$AA( " FEMF+@ FGDICF(GDIC'FEMF+*@$??@$ DPCDPCUDCUDV CUDCDCDCJ DCi DCi DV Ci DCJ DPCDPC@$!b $$==%  ;UP "#!#!;!Z!y!":"S"yS"ZS";:"#"#<>& % $$AAF`TEMF+@%@4^UU?A33@$%$$==_88833% % UP' "#!#!;!Z!y!":"S"yS"ZS";:"#"#% % $$AA( " FEMF+@ FGDICF(GDIC\'FGDICRp@"Arial0g  0 TTiK0W0 Tz08 D0DT!0Ȩm500Ȩ(T(Ȩdv%    T c|KKKAJA xLtUsually a reasonable       % ( Rp@"Arial0 " 0 TTiK0W0 Tz08 D0DT!0Ȩm500Ȩ(T(Ȩdv%    T ~KKKAJA Llguess would be:     % ( Rp@"Arial0 3T0 TTiK0W0TTz08 D0DT!0Ȩm50X Ȩ(T Ȩdv%    T`0RKKKAJA0LTlim % ( Rp@"ArialW0TL|?Q|Hm| 4??0DT!0z08$Ȩm50 Ȩ(T dv'!\P|p|m|Z|m|dv%    T`ZxKKKAJAZLTf(x % ( Rp@"ArialDL|?Q|Hm| 5??m50 Ȩ(Tz08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TdyKKKAJAyLT) =  % ( Rp@"Arial ȨL|?Q|Hm| 6??'! \P|p|m|z08<dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    T`KKKAJALTf(a % ( Rp@"Arial \L|?Q|Hm| 7??'! \P|p|m|z08Ȓdv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TXKKKAJALP)  % ( Rp@"Arial0g 0 TTiK0W0Tz08 HD0DT!0Ȩm50DȨ(TdȨdv%    TTKRKKKAJAKLPx % ( Rp@WingdingsW0L|?Q|Hm|g ??0DT!0z08Ȩm50 Ȩ(Tddv'!\P|p|m|Z|m|dv%    TTR_KKKAJARLP % ( Rp@"ArialDL|?Q|Hm|g ??m50 Ȩ(Tz08$dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TT`gKKKAJA`LPa % ( Rp@"Arial0g  0 TTiK0W0 Tz08 D0DT!0Ȩm500Ȩ(T(Ȩdv%    T KKKAJA Lx(this will be true for    % ( Rp@"Arial0(\8$0ХT8> 0z088tT8 0T0$Ȩm500Ȩ(T(Ȩdv%    T KKKAJA Lxcontinuous functions)        % ( Rp@"Arial0 U<0 TTiK0W0<Tz08\D0DT!0Ȩm50Ȩ(TxȨdv%    T| .RKKKKAJA FL\ex:  % ( Rp@"ArialW0<L|?Q|Hm| V??0DT!0z08 Ȩm50 Ȩ(Txdv'!\P|p|m|Z|m|dv%    T`S.vKKKKAJASFLTlim % ( Rp@"ArialDL|?Q|Hm| W??m50 Ȩ(Tz08$dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    T`~.KKKKAJA~FLTf(x  % ( Rp@"Arial ȨL|?Q|Hm| X??'! \P|p|m|z08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    Tp.KKKKAJAFLX) = 2  % ( Rp@"Arial0g /$0 TTiK0W0$Tz084D0DT!0Ȩm50蚥Ȩ(ThPȨdv%    TTmLt[KKKAJAmYLPx % ( Rp@WingdingsW0$L|?Q|Hm|g 0??0DT!0z08 Ȩm50 Ȩ(ThPdv'!\P|p|m|Z|m|dv%    TTtM[KKKAJAtYLP % ( Rp@"ArialDL|?Q|Hm|g 1??m50 Ȩ(Thz08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TTL[KKKAJAYLP2 % ( Rp@"Arial0g ><0 TTiK0W0<Tz08\D0DT!0Ȩm50Ȩ(TxȨdv%    T| mUKKKAJA L\but,  % ( Rp@"ArialW0<L|?Q|Hm|g ???0DT!0z08 Ȩm50 Ȩ(Txdv'!\P|p|m|Z|m|dv%    T`VmyKKKAJAVLTlim % ( Rp@"ArialDL|?Q|Hm|g @??m50 Ȩ(Tz08$dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    T`mKKKAJALTf(x % ( Rp@"Arial ȨL|?Q|Hm|g A??'! \P|p|m|z08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TpmKKKAJALX) = 7  % ( Rp@"Arial0 s$0 TTiK0W0$Tz084D0DT!0Ȩm50蚥Ȩ(ThPȨdv%    TTmtKKKAJAmLPx % ( Rp@WingdingsW0$L|?Q|Hm| t??0DT!0z08 Ȩm50 Ȩ(ThPdv'!\P|p|m|Z|m|dv%    TTtKKKAJAtLP % ( Rp@"ArialDL|?Q|Hm| u??m50 Ȩ(Thz08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TTKKKAJALP5 % ( Rp@"Arial0(\8$0ХT8> 0z088tT8 0T0$Ȩm500Ȩ(T(Ȩdv%    TwKKKAJAwLh(not f(5) = 1)     % ( Rp@"Arial0.\8$0ХT8> 0z088tT8 0T0$Ȩm50Ȩ(TxȨdv%    Tp MKKKAJA LXand  % ( Rp@"ArialL|?Q|Hm|g Z??8tT8 0T0z08 Ȩm50 Ȩ(Txdv'!\P|p|m|Z|m|dv%    T`NpKKKAJANLTlim % ( Rp@"ArialT8L|?Q|Hm|g [??m50 Ȩ(Tz08$dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    T`yKKKAJAyLTf(x % ( Rp@"Arial ȨL|?Q|Hm|g \??'! \P|p|m|z08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    T|KKKAJAL\) = DNE  % ( Rp@"Arial0 $0 TTiK0W0$Tz084D0DT!0Ȩm50蚥Ȩ(ThPȨdv%    TTipKKKAJAiLPx % ( Rp@WingdingsW0$L|?Q|Hm| ??0DT!0z08 Ȩm50 Ȩ(ThPdv'!\P|p|m|Z|m|dv%    TTp}KKKAJApLP % ( Rp@"ArialDL|?Q|Hm| ??m50 Ȩ(Thz08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TX~KKKAJA~LP16 % ( Rp@"Arial0(\8$0ХT8> 0z088tT8 0T0$Ȩm500Ȩ(T(Ȩdv%    T  KKKAJA Lx(DNE = does not exist)      % ( F(GDICV*FEMF+*@$??@&@4H@A@@@@', U]%DUBU]%DV D@'&( !b $$==_888% '%  [0V*ni)i)I7)I7)i)i)i)7)7)i)i)=i) 7) 7)=i)=i) i)c 7)c 7) i) i) i) 7) 7) i) i)W i) 7) 7)W i)W i) i)}7)}7) i) i)i)7)7)i)i)qi)97)97)qi)qi)i)7)7)i)i)-i)7)7)-i)-i)i)S7)S7)i)i)i)7)7)i)i)Gi)7)7)Gi)Gi)i)m7)m7)i)i)i)7)7)i)i)ai))7))7)ai)ai)i)7)7)i)i)i)7)7)i)i){i)C 7)C 7){i){i) i)!7)!7) i) i)7"i)z"7)z"7)7"i)7" % % $$AA( " FEMF+@ FGDICF(GDIC+FEMF+*@$??@(4(H@A33@)XLU%&DH D&DkD9~&DCUE)D:Cq ,DǙC3DC6DC@)(!b $$==_888233% % U8,)")S!)J*@*,-R% % $$AA( " FEMF+@ FGDICF(GDICbFGDICRp@"Arial0\8$0ХT8> 0z088tT8 0T0$Ȩm500Ȩ(T(Ȩdv%    TTmt KKKAJAm LP2 % ( F(GDICFGDICRp@"Arial0\8$0ХT8> 0z088tT8 0T0$Ȩm500Ȩ(T(Ȩdv%    TT KKKAJA LP5 % ( F(GDIC>FGDICRp@"Arial0\8$0ХT8> 0z088tT8 0T0$Ȩ`0ԙȨ(T(Ȩdv%    TX$2 KKKAJA$ LP10 % ( F(GDICsFGDICRp@"Arial0\8$0ХT8> 0z088tT8 0T0$Ȩ`0ԙȨ(T(Ȩdv%    TX KKKAJA LP15 % ( F(GDICUFGDICRp@"Arial0 < >0#0-0 0-z08T0I0i0-TȨdvdv@"Arialwg {Hg {dv%    T[xKKKAJAsLlOne Sided Limits % ( &% v6v% ( Rp@"Arial0l ( 0 TTiK0W0 Tz08 D0DT!0Ȩm500Ȩ(T(Ȩdv%    TKKKAJALpLimit from right:   % ( Rp@"Arial0l 5<0 TTiK0W0<Tz08 \D0DT!0Ȩm50Ȩ(TxȨdv%    T`<KKKAJALTlim % ( Rp@"ArialW0<L| Q|Hm|l 6((0DT!0z08$Ȩm50 Ȩ(Txdv'!\P|p|m|Z|m|dv%    T`DbKKKAJADLTf(x  % ( Rp@"ArialDL| Q|Hm|l 7((m50 Ȩ(Tz08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TpcKKKAJAcLX) = 5  % ( Rp@"Arial0l A<>0#0- 00-z08T0I0i0- TȨdvdv@"Arialw H dv%    TT,3KKKAJA,LPx % ( Rp@Wingdings0@|  Q|Hm| ((I0i0- Tz08  @"Arialw  dv'!PD|p|m|Z|m|dv%    TT3AKKKAJA3LP % ( Rp@"Ariali0-@|  Q|Hm| (( @"Ariaz08dv'! PD|p|m|Z|m|dv'!PD|p|m|Z|m|dv%    TXBPKKKAJABLP10 % ( Rp@"Arial @|  Q|Hm| (('! PD|p|m|z08$dv'! PD|p|m|Z|m|dv'!PD|p|m|Z|m|dv%    TTQVKKKAJAQLP+ % ( Rp@"Arial0  0 TTiK0W0 Tz08 D0DT!0Ȩm500Ȩ(T(Ȩdv%    T KKKAJALlLimit from left:  % ( Rp@"Arial0 <0 TTiK0W0<Tz08 \D0DT!0Ȩm50Ȩ(TxȨdv%    T`<8KKKAJA3LTlim % ( Rp@"ArialW0<L| Q|Hm| ((0DT!0z08$Ȩm50 Ȩ(Txdv'!\P|p|m|Z|m|dv%    T`Db8KKKAJAD3LTf(x  % ( Rp@"ArialDL| Q|Hm| ((m50 Ȩ(Tz08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    Tpc8KKKAJAc3LX) = 3  % ( Rp@"Arial0 <>0#0- 00-z08T0I0i0- TȨdvdv@"Arialwl ZHl Zdv%    TT,:3IKKKAJA,GLPx % ( Rp@Wingdings0@|  Q|Hm|l [((I0i0- Tz08  @"Arialwl Zl Zdv'!PD|p|m|Z|m|dv%    TT3;AIKKKAJA3GLP % ( Rp@"Ariali0-@|  Q|Hm|l \(( @"Ariaz08dv'! PD|p|m|Z|m|dv'!PD|p|m|Z|m|dv%    TXB:PIKKKAJABGLP10 % ( Rp@"Arial @|  Q|Hm|l ](('! PD|p|m|z08$dv'! PD|p|m|Z|m|dv'!PD|p|m|Z|m|dv%    TTQ9SDKKKAJAQCLP- % ( Rp@"Arial@`0ԧ8T,0+,08z08h8О0T @Ȩm500Ȩ(T@(Ȩdv%    TiKKKAJA~LpSince the two one     % ( Rp@"ArialL| Q|Hm|l e((О0T @z08 Ȩm50 Ȩ(T@(dv'!\P|p|m|Z|m|dv%    TTiKKKAJA~LP- % ( Rp@"Arial|/0T8\/0z088084Ȩm50ԙȨ(TȨdv%    TKKKAJALxsided limits are not       % ( Rp@"Arial0l m0 xT TiK0W0襥xTz08 D 0DT!0Ȩm50 Ȩ(T< Ȩdv%    TqKKKAJA Ldequal, then    % ( Rp@"Arial0l z<0 TTiK0W0<Tz08 \D0DT!0Ȩm50Ȩ(TxȨdv%    T`'KKKAJALTlim % ( Rp@"ArialW0<L| Q|Hm|l {((0DT!0z08$Ȩm50 Ȩ(Txdv'!\P|p|m|Z|m|dv%    T`/MKKKAJA/LTf(x  % ( Rp@"ArialDL| Q|Hm|l |((m50 Ȩ(Tz08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    T|NKKKAJANL\) = DNE  % ( Rp@"Arial0l <$>0#0- 0$0-z08T0I0i0- TȨdvdv@"Arialw H dv%    TT$KKKAJALPx % ( Rp@Wingdings0$@|  Q|Hm| ((I0i0- Tz08  @"Arialw  dv'!PD|p|m|Z|m|dv%    TT$1KKKAJA$LP % ( Rp@"Ariali0-@|  Q|Hm| (( @"Ariaz08dv'! PD|p|m|Z|m|dv'!PD|p|m|Z|m|dv%    TX2AKKKAJA2LP10 % ( F(GDIC?FGDICRp@"Arial|/0*,T8,\/0+z088084+Ȩm50ԙȨ(TȨdv%    TTF_KKKAJA[,LWhen we look at the limit below, we examine            % ( Rp@"Arial0 <<>0#0-å0<4ץ0-z08T0I0i0-åTȨdvdv@"Arialwl Hl dv%    Tda-zKKKAJAvLTthe   % ( Rp@"Arial0<@|  Q|Hm|l ((I0i0-åTz08  @"Arialwl l dv'!PD|p|m|Z|m|dv%    T`.aHzKKKAJA.vLTf(x  % ( Rp@"Ariali0-@|  Q|Hm|l (( @"Ariaz08dv'! PD|p|m|Z|m|dv'!PD|p|m|Z|m|dv%    TIazKKKAJAIvLt) values as x gets        % ( Rp@"Arial @|  Q|Hm|l (('! PD|p|m|z08$dv'! PD|p|m|Z|m|dv'!PD|p|m|Z|m|dv%    Tda4zKKKAJAvLTvery  % ( &% y65y% ( Rp@"Arial P@|  Q|Hm|l d| 0Q|Hm|l0((m|z08dv'! PD|p|m|Z| m|dv'!PD|p|m|Ztm|dv%    T<azKKKAJA<v Ldclose to a:     % ( Rp@"Arial0g <0 ЯT"TiK0W0<ЯTz08 \D#0DT!0Ȩm50Ȩ(TxȨdv%    T`KKKAJALTlim % ( Rp@"ArialW0<L| Q|Hm|g ((0DT!0z08$Ȩm50 Ȩ(Txdv'!\P|p|m|Z|m|dv%    T`KKKAJALTf(x % ( Rp@"ArialDL| Q|Hm|g ((m50 Ȩ(Tz08dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TT KKKAJALP) % ( Rp@"Arial0 +0 ,T9TiK0W0B,Tz08 HD:0DT!0Ȩm50DȨ(TdȨdv%    TTKKKAJALPx % ( Rp@WingdingsW0L| Q|Hm| ,((0DT!0z08Ȩm50 Ȩ(Tddv'!\P|p|m|Z|m|dv%    TTKKKAJALP % ( Rp@"ArialDL| Q|Hm| -((m50 Ȩ(Tz08$dv'! \P|p|m|Z|m|dv'!\P|p|m|Z|m|dv%    TTKKKAJALPa % ( F4(EMF+*@$??!b Ld)??" FEMF+@   Current User'*=GSummaryInformation(?(PowerPoint Document()+ s7DocumentSummaryInformation8D`p  Slide 1Chris HeadleeoChris Headleeo4riMicrosoft PowerPointP@&@N@0/j՜.+,0    _On-screen Show -ss7- ( / 0DArialngstt+0B 0"DTimes New Roman0B 0 DWingdingsRoman0B 0@ .  @n?" dd@  @@`` ( I 0AA@3ʚ;ʚ;g4dddd B 0HpQp? <4dddd8pC 0t+ 80___PPT10   ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> f(    6x  `}  T Click to edit Master title style! !  08{  `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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KA?HZ pC0&dT20(0Tfzd4I9 L u@ڈˊ*HqT/8 Rj ՟~t=UX@Jf#H:p!$ZZKT} ~dFɢ~$ɍ , Me MPߝa`r2s VFAcCN݌LLJ%Ӑ VyN^Dd FEb  c $A? ?3"`?2-\_ _VO{Pa%`!|-\_ _VO{P@ hJxڕRJ@=s3M0q5Th?TXhЂ7u.\s'c kM=sΜɌ 87 TOAz#(IS'ae4-EWKXC"?jMr|wGg"`+=]5ny&8:A~MwKT/2eџLݲsM[4gcJ;LpS03']:(R 90-YĚ~*0>Irg>D\_ܑ,jv3vw/ptE3Vt 7UUpFil>>q#jDd lb  c $A? ?3"`?2=UDo"%`!=UDo"J XhVxcdd``^ @b1##X`=F !#T57LSn! KA?HZ pC0&dT20(0Tfzd4I9 L u@ڈˊ2HqT/8 Rj ՟~t=UX@Juf#H:p!$ZZKT= ~dFɢ~$ɍ , Me MPߝf`r2s VFAcCN݌LLJ%Ӑ VizUiDd X/0b  c $A? ?3"`?2āö b)%`!āö bJ  7UxڕRJ@=s>-BE\ .D *(]QXE n l @>ɨHmfrs͌X OA-8"!tEP$.teA{ūPdޜXD$TUN֔g0u`+=]4nx'V[ap]U6ȐLK\fG7ie%Z36C/?/`H\?Wmow Ky:gN4*1;(p~bxTq*Ҟ{rt1,\s]}W9*J:׷[-R?K>@yYDd &&b  c $A? ?3"`?2@0#)J"b%`!w@0#)J"b @ pExcdd``ned``beV dX,XĐ Ł[Rcgb Vx Z@cjx|K2B* RA*3=S 2adBܤ&@Q meŤR@>F? )Մo UOle EPRINT 4CompObj{  l1'% EMF4\KhCF, EMF+@F\PEMF+"@ @ $@ 0@?!@ @ F(GDICFXLEMF+*@$?? @$yD;D!b $$==% % V0.x>.x>% % $$AA" FEMF+@ FGDICF(GDIC1FGDICF(GDICUFGDICF(GDICC :FGDICRp@"Arial04\4 0\[!D40PU0z08|8tP8 0P0$ܘĨ`0ЙĨ(P(Ĩdv%    TN4KKKAJAN.LxLaws of Limits Summary    % ( F(GDICIFGDICRp@"Arial0 : 0  DP`TiK0W0 [ D0PU0z08|Ka0KP!0ܘĨm50䚥Ĩ(PdPĨdv%    T!PViKKKAJA!e@LSuppose c is a constant and the and               % ( Rp@"ArialW0 L|Q|Hm| ;0KP0PU0z08Ĩm50 Ĩ(PdPdvy!\P|p|m|Z|m|dv%    TP;iKKKAJAe Ldexist. Then    % ( Rp@"ArialKL|Q|Hm| <m50 Ĩ(P0PU0z08dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    T`<PQiKKKAJA<eLT--- % ( Rp@"Arial0 A0 4DPTiK0W0`[4D0PU0z08|K0KP!0ܘĨm50,Ĩ(P(Ĩdv%    TX!2KKKAJA!LP1.  % ( Rp@"Arial0 F0 4DPTiK0W0`[4D0PU0z08|K0KP!0ܘĨm50,Ĩ(P(Ĩdv%    TX!2KKKAJA!LP2.  % ( Rp@"Arial0 K0 4DPTiK0W0`[4D0PU0z08|K0KP!0ܘĨm50,Ĩ(P(Ĩdv%    TX!2KKKAJA!LP3.  % ( Rp@"Arial0 P0 4DPTiK0W0`[4D0PU0z08|K0KP!0ܘĨm50,Ĩ(P(Ĩdv%    TX!:2SKKKAJA!OLP4.  % ( Rp@"Arial0 U0 4DPTiK0W0`[4D0PU0z08|K0KP!0ܘĨm50,Ĩ(P(Ĩdv%    TX!v2KKKAJA!LP5.  % ( Rp@"Arial0 Z0 4DPTiK0W0`[4D0PU0z08|K0KP!0ܘĨm50,Ĩ(P(Ĩdv%    TX!2KKKAJA!LP6.  % ( Rp@"Arial0 _0  DP=TiK0W0[ D0PU0z08|K>0KP!0ܘĨm50,Ĩ(P(Ĩdv%    T!KKKAJA!=L7. 8.   % ( Rp@"Arial0 d0  DP<TiK0W0[ D0PU0z08|K=0KP!0ܘĨm50,Ĩ(P(Ĩdv%    T!*CKKKAJA!?<L9. 10.   % ( Rp@"Arial0 0 8DPTiK0W0b[8D0PU0z08|K0KP!0ܘĨm50,Ĩ(P(Ĩdv%    T`!z>KKKAJA!LT11.  % ( F(GDIC}E>FGDICRp@"Arial0 u80 PDP TiK0W08n[PD0PU0z080#0- )Y0N0PU0z08T0I0i0T- )YPĨdvdvT@"Arialw H dv%    33TTQXKKKAJAQLPx % ( Rp@WingdingsY0@| Q|Hm| I0i0T- )Y0PU0z08T @"Arialw  dvy!PD|p|m|Z|m|dv%    33TTXeKKKAJAXLP % ( Rp@"Ariali0T-@| Q|Hm| T 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTfmKKKAJAfLPa % ( F(GDIC}nFGDICRp@"Arial0 80 PDP TiK0W08n[PD0PU0z080#0- Y00PU0z08T0I0i0T- YPĨdvdvT@"Arialw H dv%    33TTKKKAJALPx % ( Rp@WingdingsY0@| Q|Hm| I0i0T- Y0PU0z08T @"Arialw  dvy!PD|p|m|Z|m|dv%    33TTKKKAJALP % ( Rp@"Ariali0T-@| Q|Hm| T 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTKKKAJALPa % ( F(GDIC2/0FGDICRp@"Arial0 80 DPTiK0W08[D0PU0z080#0- Y00PU0z08T0I0i0T- YPĨdvdvT@"Arialw H dv%    33TTZ aKKKAJAZLPx % ( Rp@WingdingsY0@| Q|Hm| I0i0T- Y0PU0z08T @"Arialw  dvy!PD|p|m|Z|m|dv%    33TTa nKKKAJAaLP % ( Rp@"Ariali0T-@| Q|Hm| T 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTo vKKKAJAoLPa % ( F(GDICpFGDICRp@"Arial0  0 dDPTiK0W0 x[dD0PU0z08K0KP!0ܘĨm50䚥Ĩ(PdPĨdv%    33T`KKKAJALTlim % ( Rp@"ArialW0 L|Q|Hm| 0KP0PU0z08Ĩm50 Ĩ(PdPdvy!\P|p|m|Z|m|dv%    33T@KKKAJA L`x = a % ( Rp@"Arial0 <>0#0- D00PU0z08T0I0i0T- DPĨdvdvT@"Arialw /H /dv%    33TTKKKAJALPx % ( Rp@WingdingsD0@| Q|Hm| 0I0i0T- D0PU0z08T @"Arialw / /dvy!PD|p|m|Z|m|dv%    33TT KKKAJALP % ( Rp@"Ariali0T-@| Q|Hm| 1T 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTKKKAJALPa % ( F(GDIC/!FGDICRp@"Arial0.\4 0\[!D40PU0z088tP8 0P0$ܘĨm50+DĨ(PĨ dv%    33T`AcKKKAJAALTlim % ( Rp@"ArialDL|Q|Hm| 8tP8 0P0PU0z08Ĩm50 Ĩ(P dvy!\P|p|m|Z|m|dv%    33TTksKKKAJAkLP[  % ( Rp@"ArialP8L|Q|Hm| m50 Ĩ(P0PU0z08 dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33T`tKKKAJAtLTf(x % ( Rp@"Arial ĨL|Q|Hm| y! \P|p|0PU0z08dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TXKKKAJALP)]  % ( Rp@"Arial \L|Q|Hm| y! \P|p|0PU0z088dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TTKKKAJALP  % ( Rp@"Arial \L|Q|Hm| y! \P|p|0PU0z08Ēdvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TXKKKAJALP=  % ( Rp@"Arial0 N80 `DP TiK0W08v[`D0PU0z080#0- D00PU0z08T0I0i0T- DPĨdvdvT@"Arialw gH gdv%    33TTYG`VKKKAJAYTLPx % ( Rp@WingdingsD0@| Q|Hm| hI0i0T- D0PU0z08T @"Arialw g gdvy!PD|p|m|Z|m|dv%    33TT`HmVKKKAJA`TLP % ( Rp@"Ariali0T-@| Q|Hm| iT 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTnGuVKKKAJAnTLPa % ( F(GDICFGDICRp@"Arial0 7Ē0 \DP TiK0W0Ēt[\D0PU0z08|PK 0KP!0ܘĨm50Ĩ(PܩĨdv%    33TT07KKKAJA0LP[ % ( Rp@"ArialW0ĒL|Q|Hm| 80KP0PU0z08Ĩm50 Ĩ(Pܩdvy!\P|p|m|Z|m|dv%    33T`8ZKKKAJA8LTlim % ( Rp@"ArialKL|Q|Hm| 9m50 Ĩ(P0PU0z08 dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33T`cKKKAJAcLTf(x % ( Rp@"Arial ĨL|Q|Hm| :y! \P|p|0PU0z08dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TXKKKAJALP)]  % ( Rp@"Arial \L|Q|Hm| ;y! \P|p|0PU0z088dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TTKKKAJALP  % ( Rp@"Arial0 0 XDP TiK0W0r[XD0PU0z08(K 0KP!0ܘĨm50@Ĩ(PdĨdv%    33TTHOKKKAJAHLPx % ( Rp@WingdingsW0L|Q|Hm| 0KP0PU0z08Ĩm50 Ĩ(Pddvy!\P|p|m|Z|m|dv%    33TTO\KKKAJAOLP % ( Rp@"ArialKL|Q|Hm| m50 Ĩ(P0PU0z08 dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TT]dKKKAJA]LPa % ( F(GDIC8FFGDICRp@"Arial0 P80 PDP TiK0W08n[PD0PU0z080#0- D00PU0z08T0I0i0T- DPĨdvdvT@"Arialw H dv%    33TT]dKKKAJA]LPx % ( Rp@WingdingsD0@| Q|Hm| I0i0T- D0PU0z08T @"Arialw  dvy!PD|p|m|Z|m|dv%    33TTdqKKKAJAdLP % ( Rp@"Ariali0T-@| Q|Hm| T 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTryKKKAJArLPa % ( F(GDIC'FGDICRp@"Arial0 Ē0 \DP TiK0W0Ēt[\D0PU0z08PK 0KP!0ܘĨm50 Ĩ(Pܩ Ĩdv%    33T`9[KKKAJA9LTlim % ( Rp@"ArialW0ĒL|Q|Hm| 0KP0PU0z08 Ĩm50 Ĩ(Pܩ dvy!\P|p|m|Z|m|dv%    33T`cKKKAJAcLTf(x  % ( Rp@"ArialKL|Q|Hm| m50 Ĩ(P0PU0z08dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33T`KKKAJALT)  % ( Rp@"Arial ĨL|Q|Hm| y! \P|p|0PU0z088dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TTKKKAJALP- % ( Rp@"Arial0 <>0#0- D00PU0z08T0I0i0T- DPĨdvdvT@"Arialw H dv%    33TTQXKKKAJAQLPx % ( Rp@WingdingsD0@| Q|Hm| I0i0T- D0PU0z08T @"Arialw  dvy!PD|p|m|Z|m|dv%    33TTXeKKKAJAXLP % ( Rp@"Ariali0T-@| Q|Hm| T 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTfmKKKAJAfLPa % ( F(GDICnFGDICRp@"Arial0 80 PDP TiK0W08n[PD0PU0z080#0- D00PU0z08T0I0i0T- DPĨdvdvT@"Arialw H dv%    33TTKKKAJALPx % ( Rp@WingdingsD0@| Q|Hm| I0i0T- D0PU0z08T @"Arialw  dvy!PD|p|m|Z|m|dv%    33TTKKKAJALP % ( Rp@"Ariali0T-@| Q|Hm| T 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTKKKAJALPa % ( F(GDIC32FGDICRp@"Arial0 h0 DPTiK0W0h[D0PU0z08K0 KP!0 ܘĨm50d+DĨ(PTĨ dv%    33T`EgKKKAJAELTlim % ( Rp@"ArialW0hL|Q|Hm| 0 KP0PU0z08Ĩm50 Ĩ(PT dvy!\P|p|m|Z|m|dv%    33TTowKKKAJAoLP[  % ( Rp@"Arial KL|Q|Hm| m50 Ĩ(P0PU0z08 dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33T`xKKKAJAxLTf(x % ( Rp@"Arial ĨL|Q|Hm| y! \P|p|0PU0z08dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TXKKKAJALP)  % ( Rp@"Arial \L|Q|Hm| y! \P|p|0PU0z088dvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TTKKKAJALP- % ( Rp@"Arial \L|Q|Hm| y! \P|p|0PU0z08Pdvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33T`KKKAJALTg(x % ( Rp@"Arial \L|Q|Hm| y! \P|p|0PU0z08ܓdvy! \P|p|m|Z|m|dvy!\P|p|m|Z|m|dv%    33TpKKKAJALX)] =  % ( Rp@"Arial0 <>0#0- D00PU0z08T0I0i0T- DPĨdvdvT@"Arialw H dv%    33TT]dKKKAJA]LPx % ( Rp@WingdingsD0@| Q|Hm| I0i0T- D0PU0z08T @"Arialw  dvy!PD|p|m|Z|m|dv%    33TTdqKKKAJAdLP % ( Rp@"Ariali0T-@| Q|Hm| T 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTryKKKAJArLPa % ( F(GDIC0FGDICRp@"Arial0 80 XDP TiK0W08r[XD0PU0z08|0#0- D00PU0z08T0I0i0T- DPĨdvdvT@"Arialw EH Edv%    33TTVeKKKAJAcLPx % ( Rp@WingdingsD0@| Q|Hm| FI0i0T- D0PU0z08T @"Arialw E Edvy!PD|p|m|Z|m|dv%    33TTWeKKKAJAcLP % ( Rp@"Ariali0T-@| Q|Hm| GT 0PU0z08 dvy! PD|p|m|Z|m|dvy!PD|p|m|Z|m|dv%    33TTVeKKKAJAcLPa % ( F(GDIC110lFGDICRp@"Arial0 *XN0 NDPTiK0W0XN[D0PU0z08K0 KP!0 ܘĨ m50,DĨ( P0KĨ dv%    33T`C7eTKKKAJACOLTlim % ( Rp@"ArialW0XNL|Q|Hm| +0 KP0PU0z08Ĩm50 Ĩ( P0K dv y!\P|p|m|Z|m|dv%    33TTm7uTKKKAJAmOLP[  % ( Rp@"Arial KL|Q|Hm| ,m50 Ĩ( P0PU0z08 dvy! \P|p|m|Z|m|dv y!\P|p|m|Z|m|dv%    33T`v7TKKKAJAvOLTf(x % ( Rp@"Arial ĨL|Q|Hm| -y! \P|p|0PU0z08dvy! \P|p|m|Z|m|dv y!\P|p|m|Z|m|dv%    33TX7TKKKAJAOLP)  % ( Rp@"Arial \L|Q|Hm| .y! \P|p|0PU0z088dvy! \P|p|m|Z|m|dv y!\P|p|m|Z|m|dv%    33TT7TKKKAJAOLP"  % ( Rp@"Arial \L|Q|Hm| /y! \P|p|0PU0z08Pdvy! \P|p|m|Z|m|dv y!\P|p|m|Z|m|dv%    33T`7TKKKAJAOLTg(x % ( Rp@"Arial \L|Q|Hm| 0y! \P|p|0PU0z08ܓdvy! \P|p|m|Z|m|dv y!\P|p|m|Z|m|dv%    33Tl7TKKKAJAOLX)]  % ( Rp@"Arial \L|Q|Hm| 1y! \P|p|0PU0z08Ndvy! \P|p|m|Z|m|dv y!\P|p|m|Z|m|dv%    33TT7TKKKAJAOLP= % ( Rp@"Arial0 ;<>0#0- 00PU0z08T0I0i0T- PĨdvdvT@"Arialw hH hdv%    33TT[VbeKKKAJA[cLPx % ( Rp@Wingdings0@| Q|Hm| iI0i0T- 0PU0z08T @"Arialw h hdvy!PD|p|m|Z|m|dv%    33TTbWoeKKKAJAbcLP % ( Rp@"Ariali0T-@| Q|Hm| jT 0PU0z08 dvy! 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KKKAJA!LlSQUEEZE THEOREM  % ( &% !6% ( Rp@"ArialK0@| Q|Hm| d|0Q|Hm|00PU0z08T @"Arialw  dv!PD|p|m|Ztm|dv%    TT KKKAJALP: % ( Rp@"Arial0 KV0 V DPVTiK0W0KV[ D0PU0z08|HHNW0 LGNP!0 ܘĨ m50,DĨ( P0KĨ dv%    T!&?KKKAJA!;L|(Conditions:) 1) If     % ( Rp@"ArialW0KVL|Q|Hm| 0 LGNP0PU0z08Ĩm50 Ĩ( P0K dv !\P|p|m|Z|m|dv%    T`&?KKKAJA;LTf(x  % ( Rp@"Arial LGNL|Q|Hm| m50 Ĩ( P0PU0z08dv! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TX& ?KKKAJA;LP)  % ( Rp@"Arial ĨL|Q|Hm| ! \P|p|0PU0z08 dv! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TT!&,?KKKAJA!;LPd"  % ( Rp@"Arial \L|Q|Hm| ! \P|p|0PU0z088dv! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`4&T?KKKAJA4;LTg(x   % ( Rp@"Arial \L|Q|Hm| ! \P|p|0PU0z08Ēdv! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TXU&a?KKKAJAU;LP)  % ( Rp@"Arial \L|Q|Hm| ! \P|p|0PU0z08Pdv! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    TTb&m?KKKAJAb;LPd"  % ( Rp@"Arial \L|Q|Hm| ! \P|p|0PU0z08hdv! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T`u&?KKKAJAu;LTh(x   % ( Rp@"Arial \L|Q|Hm| ! \P|p|0PU0z08KVdv! \P|p|m|Z|m|dv !\P|p|m|Z|m|dv%    T\&e?KKKAJA;-L) when x is near a (except possibly at a) and             % ( Rp@"Arial0 0 DPTiK0W0[D0PU0z08|GN0LGNP!0ܘĨm50,Ĩ(P(Ĩdv%    TX[tKKKAJApLP2)  % ( Rp@"Arial0q w0 <DPTiK0W0d[<D0PU0z08|GN0LGNP!0ܘĨm50,Ĩ(P(Ĩdv%    Td!NKKKAJA!LTthen  % ( F(GDIC\FGDICRp@"Arial0 0Ē0 pDPTiK0W0Ē~[pD0PU0z08GN0LGNP!0ܘĨm50Ĩ(PܩĨdv%    33T`nKKKAJAnLTlim % ( Rp@"ArialW0ĒL|Q|Hm| 10LGNP0PU0z08 Ĩm50 Ĩ(Pܩdv!\P|p|m|Z|m|dv%    33T`KKKAJALTf(x  % ( Rp@"ArialLGNL|Q|Hm| 2m50 Ĩ(P0PU0z08dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33T|KKKAJAL\)  % ( Rp@"Arial ĨL|Q|Hm| 3! \P|p|0PU0z088dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33TTKKKAJALPd" % ( Rp@"Arial0q 0 XDP TiK0W0r[XD0PU0z08GN 0LGNP!0ܘĨm50@Ĩ(PdĨdv%    33TTKKKAJALPx % ( Rp@WingdingsW0L|Q|Hm|q 0LGNP0PU0z08Ĩm50 Ĩ(Pddv!\P|p|m|Z|m|dv%    33TTKKKAJALP % ( Rp@"ArialLGNL|Q|Hm|q m50 Ĩ(P0PU0z08 dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33TTKKKAJALPa % ( F(GDICFGDICRp@"Arial0 H80 PDP TiK0W08n[PD0PU0z08GN 0LGNP!0ܘĨm50 Ĩ(Px Ĩdv%    33T`(JKKKAJA(LTlim % ( Rp@"ArialW08L|Q|Hm| I0LGNP0PU0z08 Ĩm50 Ĩ(Px dv!\P|p|m|Z|m|dv%    33T`RwKKKAJARLTg(x % ( Rp@"ArialLGNL|Q|Hm| Jm50 Ĩ(P0PU0z08dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33TTxKKKAJAxLP) % ( Rp@"Arial0q 0 XDP TiK0W0r[XD0PU0z08GN 0LGNP!0ܘĨm50@Ĩ(PdĨdv%    33TT@GKKKAJA@LPx % ( Rp@WingdingsW0L|Q|Hm|q 0LGNP0PU0z08Ĩm50 Ĩ(Pddv!\P|p|m|Z|m|dv%    33TTGTKKKAJAGLP % ( Rp@"ArialLGNL|Q|Hm|q m50 Ĩ(P0PU0z08 dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33TTU\KKKAJAULPa % ( F(GDICUFGDICRp@"Arial0 aĒ0 pDPTiK0W0Ē~[pD0PU0z08GN0LGNP!0ܘĨm50Ĩ(PܩĨdv%    T`[xKKKAJAsLTlim % ( Rp@"ArialW0ĒL|Q|Hm| b0LGNP0PU0z08 Ĩm50 Ĩ(Pܩdv!\P|p|m|Z|m|dv%    T`[.xKKKAJAsLTf(x  % ( Rp@"ArialLGNL|Q|Hm| cm50 Ĩ(P0PU0z08dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    T|/[gxKKKAJA/sL\)  % ( Rp@"Arial ĨL|Q|Hm| d! \P|p|0PU0z088dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TTh[vxKKKAJAhsLP= % ( Rp@"Arial0q 0 XDP TiK0W0r[XD0PU0z08GN 0LGNP!0ܘĨm50@Ĩ(PdĨdv%    TTyKKKAJALPx % ( Rp@WingdingsW0L|Q|Hm|q 0LGNP0PU0z08Ĩm50 Ĩ(Pddv!\P|p|m|Z|m|dv%    TTzKKKAJALP % ( Rp@"ArialLGNL|Q|Hm|q m50 Ĩ(P0PU0z08 dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TTyKKKAJALPa % ( F(GDICVqFGDICRp@"Arial0 y80 DPTiK0W08[D0PU0z08GN0LGNP!0ܘĨm50Ĩ(PxĨdv%    T`\yKKKAJAtLTlim % ( Rp@"ArialW08L|Q|Hm| z0LGNP0PU0z08 Ĩm50 Ĩ(Pxdv!\P|p|m|Z|m|dv%    T`\yKKKAJAtLTh(x % ( Rp@"ArialLGNL|Q|Hm| {m50 Ĩ(P0PU0z08dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    T\\yKKKAJAt Lh) = L % ( Rp@"Arial0q 0 XDP TiK0W0r[XD0PU0z08GN 0LGNP!0ܘĨm50@Ĩ(PdĨdv%    TTzKKKAJALPx % ( Rp@WingdingsW0L|Q|Hm|q 0LGNP0PU0z08Ĩm50 Ĩ(Pddv!\P|p|m|Z|m|dv%    TT{KKKAJALP % ( Rp@"ArialLGNL|Q|Hm|q m50 Ĩ(P0PU0z08 dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    TTzKKKAJALPa % ( F(GDICFGDICRp@"Arial0 80 `DP TiK0W08v[`D0PU0z08GN0LGNP!0ܘĨm50 Ĩ(Px Ĩdv%    33T`"DKKKAJA"LTlim % ( Rp@"ArialW08L|Q|Hm| 0LGNP0PU0z08 Ĩm50 Ĩ(Px dv!\P|p|m|Z|m|dv%    33T`LqKKKAJALLTg(x % ( Rp@"ArialLGNL|Q|Hm| m50 Ĩ(P0PU0z08dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33TlrKKKAJArLX) = L % ( Rp@"Arial0q 0 XDP TiK0W0r[XD0PU0z08GN 0LGNP!0ܘĨm50@Ĩ(PdĨdv%    33TT:AKKKAJA:LPx % ( Rp@WingdingsW0L|Q|Hm|q 0LGNP0PU0z08Ĩm50 Ĩ(Pddv!\P|p|m|Z|m|dv%    33TTANKKKAJAALP % ( Rp@"ArialLGNL|Q|Hm|q m50 Ĩ(P0PU0z08 dv! \P|p|m|Z|m|dv!\P|p|m|Z|m|dv%    33TTOVKKKAJAOLPa % ( F(GDIC6CF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, gC9rDxD/D@( !b $$==_888% '%  ;e#6=#X4=#=#=#=#=#=#6e#X4_#Z#Z#Z#_#e#==#Y$=#=#=<?7A % % $$AA( " FEMF+@ FGDICF(GDICuF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, s>D;Ds>DC@!b $$==_888% %  ;/.6/X4//////6/.X4/./././././.=g/Y$/h/=<?u % % $$AA( " FEMF+@ FGDICF(GDICzn?FEMF+*@$??@@4UU@AA@@@XLDbCU)DCl4DPDs>D/D9zHD9DQDrC$[DrC@( !b $$==_888% '%  [xo?  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FEMF+@ Current UserGSummaryInformation((PowerPoint Document(~ !DocumentSummaryInformation8Slide 1Chris HeadleeoChris Headleeo9riMicrosoft PowerPointP@RG @N@{U՜.+,0    KOn-screen Show -s!:  N ! " ( `/ 0DArialngstt+0B 0"DWingdingstt+0B 0@ .  @n?" dd@  @@`` (  0AA @3ʚ;ʚ;g4\d\d B 0p2p? <4dddd8pC 0t+ 80___PPT10   ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> f(    6x  `}  T Click to edit Master title style! !  04{  `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0 ^ `  >*  0 ^   @*  0l ^ `  @*H  0޽h ? 3380___PPT10.K tv Default Design <4 (   E <0OX&I ASqueeze Theorem>  6tQ  Theorem: If f(x) d" g(x) when x is near a (except possibly at a) and the limits of f and g exist as x approaches a, then SQUEEZE THEOREM: (Conditions:) 1) If f(x) d" g(x) d" h(x) when x is near a (except possibly at a) and 2) then@ k""~"b RL  6pХ9$  : lim f(x) d" xaJ$   >,  6<Y? V * lim g(x) xa:    > <  6(oK  : lim f(x) = xa:    >8  6rK  B lim h(x) = L xa."   >4  6N "l  2 lim g(x) = L xa:   >RB  s *D T RB  s *Dp$$  # BlCDEFAjJl@  S"  M   3 BlCDEFjJl@  S"  nh  @B6CDEF@AA((Mm` 5.Z&uS1%#12?LM`s %=0u?N_;s2<'6 @             ' u   <4N   Hy = x&$  <hMN Iy = -x&$  <HN   Jy = x sin (1/x)    <p;N| Q P  FFigure 8 from pg 111  <,UKVD  3y  <QN K  3xH  0޽h ? 3380___PPT10.K tvr0w eK!   !#$%&'(),/256789<?BEFGJMOPQRTUVWXYZ]`bcdeghijklmpstwz{~Arial WingdingsDefault DesignSlide 1  Fonts UsedDesign Template Slide Titles FMicrosoft Equation 3.0 DS Equation Equation.39qlII 'lim x!_1118918553FICUICUOle CompObjfObjInfoEquation Native _1118918571FICUICUOle  CompObj f6 4x 2 "10x+3() FMicrosoft Equation 3.0 DS Equation Equation.39qlII 'lim x!5  4x+446x"29 ObjInfo Equation Native _1216455341?5FICUICUOle 3 FMicrosoft Equation 3.0 DS Equation Equation.39qD\ 'lim x!3 fx()CompObjfObjInfoEquation Native `_1246876169{dO)ICUEUOle EPRINT CompObj{ObjInfo                          # 8 9 % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 : ; < = > @ ? A B E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  l1'% EMF\KhCF, EMF+@F\PEMF+"@ @ $@ 0@?!@ @ F(GDICFXLEMF+*@$?? @$yD;D!b $$=='% % V0.x>.x>% % $$AA" FEMF+@ FGDICF(GDIC1FGDICF(GDICUFGDICF(GDIC| m:FGDICRp@"Arial04\4 0\[!D40PU0z08|8tP8 0P0$ܘĨ`0Ĩ(PdPĨdv%    Tp4KKKAJA.LXLimit   % ( Rp@"ArialDL|}Q|Hm|] |}}8tP8 0P0PU0z08Ĩ`0 Ĩ(PdPdv !x\P|p|m|Z|m|dv%    TT4KKKAJA.LP-  % ( Rp@"ArialP8L|}Q|Hm|] }}}`0 Ĩ(P0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Ta4KKKAJA. L`Definition   % ( F(GDICIQFGDICRp@"Arial 0PU0z08|`D0@0Ч4PP820HY4Чdv%    T!PiKKKAJA!e L`Definition   % ( &% !h6h% ( Rp@"Arial@|} Q|Hm|} d|}0Q|Hm|}0}}0PU0z08P820 Y4 dvP !xPD|p|m|Ztm|dv%    TPiKKKAJAe@L: Let f be a function defined on some open interval that contai                  % ( Rp@"ArialQ|Hm|@|} Q|Hm| )}}820 Y0PU0z08dv !x PD|p|m|Zm|dvP !xPD|p|m|Z|m|dv%    TPiKKKAJAeLhns the number     % ( Rp@"Arial0 @|} Q|Hm| *}} !x PD|p|0PU0z08 dv !x PD|p|m|Z|m|dvP !xPD|p|m|Z|m|dv%    TTQiKKKAJAeLPa  % ( Rp@"Arial P@|} Q|Hm| +}} !x PD|p|0PU0z08dv !x PD|p|m|Z|m|dvP !xPD|p|m|Z|m|dv%    TXPiKKKAJAeLP,  % ( Rp@"Arial|/0[]P4]\/00PU0z08|4044\ܘĨm50@5Ĩ(Pd5Ĩdv%    T!kKKKAJA!Ltexcept possibly at      % ( Rp@"Arial]L|}Q|Hm|] }}044\0PU0z08Ĩm50 5Ĩ(Pd5dv !x\P|p|m|Z|m|dv%    TTlKKKAJALPa  % ( Rp@"Arial044L|}Q|Hm|] }}m50 5Ĩ(P0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    T8kpKKKAJA'Litself. Then we say that the limit of        % ( Rp@"Arial 5ĨL|}Q|Hm|] }} !x \P|p|0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    T`qkKKKAJAqLTf(x  % ( Rp@"Arial \L|}Q|Hm|] }} !x \P|p|0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TkKKKAJAL) as x approaches a is L, and           % ( Rp@"Arial0 90 PDP TiK0W0n[PD0PU0z08|GN 0LGNP!0ܘĨm50,Ĩ(P(Ĩdv%    T!}KKKAJA! L`we write    % ( Rp@"Arial0] P0  DP<TiK0W0P[ D0PU0z08|GN=0LGNP!0ܘĨm50TĨ(PĨdv%    T!KKKAJA!Ltif for every number     % ( Rp@"ArialW0PL|}Q|Hm|] }}0LGNP0PU0z08Ĩm50 Ĩ(Pdv !x\P|p|m|Z|m|dv%    Rp@"ArialPL|}Q|Hm|] }}0LGNP0PU0z08Ĩww!x4 0w !xK :  /T  w !x(wI  dv% %  TTKKKAJALP  % % ( h ( Rp@"ArialLGNL|}Q|Hm|] }}m50 Ĩ(P0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TKKKAJAL|> 0 there is a number          ( Rp@"Arial ĨL|}Q|Hm|] }} !x \P|p|0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Rp@"ArialĨL|}Q|Hm|] }} !x \P|p|0PU0z08dvww!x4 0w !xK :  /T  w !x(wXP  dv% %  TTKKKAJALP  % % ( h ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z08Ēdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TKKKAJA Lh> 0 such that      ( Rp@"Arial0} N0N DP=TiK0W0N[ D0PU0z08\HN>0 LGNP!0 ܘĨ m50x,DĨ( P1KĨ dv%    TTk q$KKKAJAk LP|& ( Rp@"ArialW0NL|}Q|Hm|} }}0 LGNP0PU0z08Ĩm50 Ĩ( P1K dv !x\P|p|m|Z|m|dv%    T`r $KKKAJAr LTf(x  ( Rp@"Arial LGNL|}Q|Hm| }}m50 Ĩ( P0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TX $KKKAJA LP)  ( Rp@"Arial ĨL|}Q|Hm|} }} !x \P|p|0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TT $KKKAJA LP   ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z088dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Tl $KKKAJA LXL| <   ( Rp@"Arial \L|}Q|Hm|} }} !x \P|p|0PU0z08Ēdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%     Rp@"ArialL|}Q|Hm|} }} !x \P|p|0PU0z08Ēdvww!x4 0w !xK :  /T  w !x(w dv% %  TT $KKKAJA LP  % % ( h ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z08Pdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    T $KKKAJA  Lhwhenever    ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z08ܓdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Tx $KKKAJA L\0 < |x    ( Rp@"Arial \L|}Q|Hm|] }} !x \P|p|0PU0z08hdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TT $KKKAJA LP  ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z084Ndv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Tl $KKKAJA LXa| <   ( Rp@"Arial \L|}Q|Hm|] }} !x \P|p|0PU0z08Ndv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%     Rp@"ArialL|}Q|Hm|] }} !x \P|p|0PU0z08Ndvww!x4 0w !xK :  /T  w !x(w dv% %  TT  $KKKAJA LP  % % ( h ( Rp@"Arial0 0 \DP TiK0W0t[\D0PU0z08|GN 0LGNP!0ܘĨm50,Ĩ(P(Ĩdv%    T!AZKKKAJA!V LdAnother way:    ( Rp@"Arial0} 8WN0 N DP5TiK0W0WN[ D0PU0z08|HN60LGNP!0ܘĨ m500-DĨ( P01K,Ĩdv%    T`lv~KKKAJAlLTif  ( Rp@"ArialW0WNL|}Q|Hm|} 9}}0LGNP0PU0z08Ĩm50 Ĩ( P01K,dv !x\P|p|m|Z|m|dv%    TxvKKKAJAL\0 < |x    ( Rp@"ArialLGNL|}Q|Hm| }}m50 Ĩ( P0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TTvKKKAJALP  ( Rp@"Arial ĨL|}Q|Hm|} ;}} !x \P|p|0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TlvKKKAJALXa| <   ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Rp@"ArialL|}Q|Hm| }} !x \P|p|0PU0z08dvww!x4 0w !xK :  /T  w !x(w`  dv% %  TTv KKKAJALP  % % ( h ( Rp@"Arial \L|}Q|Hm|} =}} !x \P|p|0PU0z088dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TlvKKKKAJALXthen   ( Rp@"Arial \L|}Q|Hm|} >}} !x \P|p|0PU0z08Pdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TTRvWKKKAJARLP| ( Rp@"Arial \L|}Q|Hm|] }} !x \P|p|0PU0z08ܓdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    T`XvsKKKAJAXLTf(x  ( Rp@"Arial \L|}Q|Hm|} @}} !x \P|p|0PU0z08hdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TXtvKKKAJAtLP)  ( Rp@"Arial \L|}Q|Hm|] }} !x \P|p|0PU0z08Ndv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TTvKKKAJALP  ( Rp@"Arial \L|}Q|Hm|} B}} !x \P|p|0PU0z08Ndv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TlvKKKAJALXL| <   ( Rp@"Arial \L|}Q|Hm|] }} !x \P|p|0PU0z084Ndv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Rp@"ArialL|}Q|Hm|] }} !x \P|p|0PU0z084Ndvww!x4 0w !xK :  /T  w !x(w dv% %  TTvKKKAJALP  % % ( h ( Rp@"Arial0} H0 dDPTiK0W0x[dD0PU0z08|GN0LGNP!0ܘĨm50,Ĩ(P(Ĩdv%    T!KKKAJA!LhAlternatively:     ( Rp@"Arial0  0 , DP@TiK0W0 [, D0PU0z08|GNA0LGNP!0ܘĨm50䚥Ĩ(PdPĨdv%    TKKKAJALxmeans that for every      ( Rp@"ArialW0 L|}Q|Hm| }}0LGNP0PU0z08Ĩm50 Ĩ(PdPdv !x\P|p|m|Z|m|dv%    Rp@"Arial L|}Q|Hm| }}0LGNP0PU0z08Ĩww!x4 0w !xK :  /T  w !x(w% dv% %  TTKKKAJALP  % % ( h ( Rp@"ArialLGNL|}Q|Hm|} K}}m50 Ĩ(P0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TdKKKAJALT> 0   ( Rp@"Arial0  0 DP2TiK0W0 [D0PU0z08|GN30LGNP!0ܘĨm50䚥Ĩ(PdPĨdv%    T!KKKAJA!"L(no matter how small) we can find           ( Rp@"ArialW0 L|}Q|Hm| }}0LGNP0PU0z08Ĩm50 Ĩ(PdPdv !x\P|p|m|Z|m|dv%    Rp@"Arial L|}Q|Hm| }}0LGNP0PU0z08Ĩww!x4 0w !xK :  /T  w !x(w. dv% %  TTKKKAJALP % % ( h ( Rp@"ArialLGNL|}Q|Hm|} N}}m50 Ĩ(P0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    T.KKKAJALh> 0 such that      ( Rp@"Arial0] h0  DP9TiK0W0h[ D0PU0z08| HN:0 LGNP!0 ܘĨm50d+DĨ(PTĨ dv%    T!^/KKKAJA!+"Lif x lies in the open interval (a           ( Rp@"ArialW0hL|}Q|Hm|] }}0 LGNP0PU0z08Ĩm50 Ĩ(PT dv !x\P|p|m|Z|m|dv%    TT_e/KKKAJA_+LP- ( Rp@"Arial LGNL|}Q|Hm|} Q}}m50 Ĩ(P0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    l+Rp@"ArialLGNL|}Q|Hm|} Q}}m50 Ĩ(P0PU0z08 dvww!x4 0w !xK :  /T  w !x(w9 dv% %  TTly/KKKAJAl+LP % % ( h ( Rp@"Arial ĨL|}Q|Hm|] }} !x \P|p|0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Tpz/KKKAJAz+LX, a +    ( Rp@"Arial \L|}Q|Hm|] }} !x \P|p|0PU0z088dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    +Rp@"ArialL|}Q|Hm|] }} !x \P|p|0PU0z088dvww!x4 0w !xK :  /T  w !x(w? dv% %  TT/KKKAJA+LP % % ( h ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z08Ēdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    T|/KKKAJA+L\) and x     ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z08Pdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TT/KKKAJA+LP`"  ( Rp@"Arial \L|}Q|Hm|} U}} !x \P|p|0PU0z08ܓdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    T`2/KKKAJA+LTa,  ( Rp@"Arial0 h0 DP2TiK0W0h[D0PU0z08| HN30 LGNP!0 ܘĨm50d+DĨ(PTĨ dv%    Tl!1UJKKKAJA!FLXthen   ( Rp@"ArialW0hL|}Q|Hm| }}0 LGNP0PU0z08Ĩm50 Ĩ(PT dv !x\P|p|m|Z|m|dv%    T`V1pJKKKAJAVFLTf(x  ( Rp@"Arial LGNL|}Q|Hm| }}m50 Ĩ(P0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Tq1JKKKAJAqFL) lies in the open interval (L         ( Rp@"Arial ĨL|}Q|Hm| }} !x \P|p|0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TT1JKKKAJAFLP  ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z088dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    FRp@"ArialL|}Q|Hm| }} !x \P|p|0PU0z088dvww!x4 0w !xK :  /T  w !x(w S dv% %  TT1JKKKAJAFLP  % % ( h ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z08Ēdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Tp1JKKKAJAFLX, L +   ( Rp@"Arial \L|}Q|Hm| }} !x \P|p|0PU0z08Pdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    FRp@"ArialL|}Q|Hm| }} !x \P|p|0PU0z08Pdvww!x4 0w !xK :  /T  w !x(w$Y dv% %  TT1JKKKAJAFLP  % % ( h ( Rp@"Arial \L|}Q|Hm|] }} !x \P|p|0PU0z08ܓdv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    TT1JKKKAJAFLP) ( F(GDIC4FGDICRp@"Arial0} t80 DPTiK0W08[D0PU0z08GN0LGNP!0ܘĨm50Ĩ(PxĨdv%    33T`FhKKKAJAFLTlim ( Rp@"ArialW08L|}Q|Hm|} u}}0LGNP0PU0z08 Ĩm50 Ĩ(Pxdv !x\P|p|m|Z|m|dv%    33T`pKKKAJApLTf(x  ( Rp@"ArialLGNL|}Q|Hm|] }}m50 Ĩ(P0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    33TKKKAJA Lh) = L ( Rp@"Arial0 0 XDP TiK0W0r[XD0PU0z08GN 0LGNP!0ܘĨm50@Ĩ(PdĨdv%    33TT^eKKKAJA^LPx ( Rp@WingdingsW0L|}Q|Hm| }}0LGNP0PU0z08Ĩm50 Ĩ(Pddv !x\P|p|m|Z|m|dv%    33TTerKKKAJAeLP ( Rp@"ArialLGNL|}Q|Hm|} y}}m50 Ĩ(P0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    33TTszKKKAJAsLPa ( F(GDICygF(GDIC%2F EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, "DU DzDU D@( !b $$==_888% '%  ;("6+>"X41>"6>"6>"6>"1>"+>"6("X4("("("("("("=>~"Y$x>">"=<?&0 % % $$AA( " FEMF+@ FGDICF(GDIC`F EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, 0DD0DaC@!b $$==_888% %  ;%,%6%,<X4%,6*,10,16,1;,6;,<6;,%X4;,%6,%0,%*,%%,%%,%=,KY$0,^,K=<?a % % $$AA( " FEMF+@ FGDICF(GDICFGDICRp@"Arial0#\4 0\[!D40PU0z08|8tP8 0P0$ܘĨm50,Ĩ(P(Ĩdv%    TTKKKAJALPy  ( F(GDIC%IFGDICRp@"Arial0#\4 0\[!D40PU0z08|8tP8 0P0$ܘĨm50,Ĩ(P(Ĩdv%    TT,AKKKAJA>LPx  ( F(GDICgFEMF+*@$??@4(UU@A@XL*DWD6D~ DBDaCKDoCrUDr|C[DHC bD*~C@!b $$==_888!% % U8g*N&-#02&=5h6"{8% % $$AA( " FEMF+@ FGDICF(GDIC9;+FEMF+*@$??@@4^UU?A@, wND DwNDsC@!b $$==_888% % W$7;,3"3% % $$AA( " FEMF+@ FGDICF(GDICm@o*FEMF+*@$??@@4^UU?A@ , q[D& Dq[D䀠C@ !b $$==_888% % W$k?o*6"6 % % $$AA( " FEMF+@ FGDICF(GDIC@nBFEMF+*@$??@ @4^UU?A@ , 9:[DUC0DUC@  !b $$==_888% % W$?nC6,% % $$AA( " FEMF+@ FGDICF(GDIC9FEMF+*@$??@ @4^UU?A@ , NDC0DC@  !b $$==_888% % W$:}35,5% % $$AA( " FEMF+@ FGDICF(GDICNxZFEMF+*@$??@ UDCiTDCUSD-CUSDCUSD sCiTDCUDCUDC]VD sC]VDC]VD-CUDCUDC@( !b $$=='%  ;UP ?554445+?5+l5+555l5?5<>NxY % $$AAF`TEMF+@@4^UU?A@$$==_888% % UPMw[ ?554445+?5+l5+555l5?5% % $$AA( " FEMF+@ FGDICF(GDICxFEMF+*@$??@ 0D9^C60D9^C/D~C/DC/DFC60DgC0DgCܚ1DgC+2DFC+2DC+2D~Cܚ1D9^C0D9^C@( !b $$=='%  ;UP 3,,+++,%3,%_,%,,,_,3,<>x % $$AAF`TEMF+@@4^UU?A@$$==_888% % UPv 3,,+++,%3,%_,%,,,_,3,% % $$AA( " FEMF+@ FGDICF(GDICP$\0FEMF+*@$??@ rtUDS DzTDS D92TD D92TDǕ D92TDG DzTD DrtUD Dj&VD DVDG DVDǕ DVD Dj&VDS DrtUDS D@!b $$==%  ;UP V5M")5M"5q"5"5")5"V5"5"5"5"5q"5M"V5M"<>P$[/ % $$AAF`TEMF+@@4^UU?A@$$==_888% % UPN#\1 V5M")5M"5q"5"5")5"V5"5"5"5"5q"5M"V5M"% % $$AA( " FEMF+@ FGDICF(GDICNZFEMF+*@$??@ UUD䀇C]STD䀇CSDfCSDVCSDEiC]STDȉCUUDȉCMUDȉCGVDEiCGVDVCGVDfCMUD䀇CUUD䀇C@( !b $$=='%  ;UP :5 54 494f 5:5f55f595 f5:5<>NY % $$AAF`TEMF+@@4^UU?A@$$==_888% % UPM [ :5 54 494f 5:5f55f595 f5:5% % $$AA( " FEMF+@ FGDICF(GDICF*gNFGDICRp@"Arial0#\4 0\[!D40PU0z08|8tP8 0P0$ܘĨm50,Ĩ(P(Ĩdv%    TTQ1[FKKKAJAQCLPa  ( F(GDIC/:RFGDICRp@"Arial0#\4 0\[!D40PU0z08|8tP8 0P0$ܘĨm50@Ĩ(PdĨdv%    TX5JKKKAJAGLPa  ( Rp@"ArialDL|Q|Hm|] 8tP8 0P0PU0z08Ĩm50 Ĩ(Pddv !x\P|p|m|Z|m|dv%    TT5JKKKAJAGLP- ( Rp@"ArialP8L|Q|Hm| Иm50 Ĩ(P0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    "GRp@"Arial8L|Q|Hm| Иm50 Ĩ(P0PU0z08 dvww!x4 0w !xK 2  /T  w !x(w dv% %  TT"5-JKKKAJA"GLP  % % ( h ( F(GDICl/RFGDICRp@"Arial0#\4 0\[!D40PU0z08|8tP8 0P0$ܘĨm50Ĩ(P<<Ĩdv%    Tdw5JKKKAJAwGLTa +   ( Rp@"ArialDL|Q|Hm|} 8tP8 0P0PU0z08Ĩm50 Ĩ(P<<dv !x\P|p|m|Z|m|dv%    GRp@"ArialL|Q|Hm|} 8tP8 0P0PU0z08Ĩww!x4 0w !xK 2  /T  w !x(w dv% %  TT5JKKKAJAGLP  % % ( h ( F(GDICxFGDICRp@"Arial0#\4 0\[!D40PU0z08|8tP8 0P0$ܘĨm50@Ĩ(PdĨdv%    TXKKKAJALPL  ( Rp@"ArialDL|Q|Hm|] 8tP8 0P0PU0z08Ĩm50 Ĩ(Pddv !x\P|p|m|Z|m|dv%    TTKKKAJALP- ( Rp@"ArialP8L|Q|Hm| Ԙm50 Ĩ(P0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    Rp@"Arial8L|Q|Hm| Ԙm50 Ĩ(P0PU0z08 dvww!x4 0w !xK 2  /T  w !x(w0 dv% %  TTKKKAJALP  % % ( h ( F(GDICx4WFGDICRp@"Arial0#\4 0\[!D40PU0z08|8tP8 0P0$ܘĨm50Ĩ(P<<Ĩdv%    Td:OKKKAJALLTL +   ( Rp@"ArialDL|Q|Hm| 18tP8 0P0PU0z08Ĩm50 Ĩ(P<<dv !x\P|p|m|Z|m|dv%    LRp@"ArialL|Q|Hm| 18tP8 0P0PU0z08Ĩww!x4 0w !xK 2  /T  w !x(wx dv% %  TT:OKKKAJALLP  % % ( h ( F(GDIClFGDICRp@"Arial0#\4 0\[!D40PU0z08|8tP8 0P0$ܘĨm50,Ĩ(P(Ĩdv%    TTrKKKAJALPL  ( FGDICF(GDICFGDICRp@"Arial0} 80 DPTiK0W08[D0PU0z08GN0LGNP!0ܘĨm50Ĩ(PxĨdv%    33T`%GKKKAJA%LTlim ( Rp@"ArialW08L|Q|Hm|} 0LGNP0PU0z08 Ĩm50 Ĩ(Pxdv !x\P|p|m|Z|m|dv%    33T`OmKKKAJAOLTf(x  ( Rp@"ArialLGNL|Q|Hm|] *m50 Ĩ(P0PU0z08dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    33TnKKKAJAn Lh) = L ( Rp@"Arial0 0 XDP TiK0W0r[XD0PU0z08GN 0LGNP!0ܘĨm50@Ĩ(PdĨdv%    33TT=DKKKAJA=LPx ( Rp@WingdingsW0L|Q|Hm| 영0LGNP0PU0z08Ĩm50 Ĩ(Pddv !x\P|p|m|Z|m|dv%    33TTDQKKKAJADLP ( Rp@"ArialLGNL|Q|Hm|} m50 Ĩ(P0PU0z08 dv !x \P|p|m|Z|m|dv !x\P|p|m|Z|m|dv%    33TTRYKKKAJARLPa ( F4(EMF+*@$??!b Ld)??" FEMF+@  dO)"Microsoft PowerPoint PresentationMSPresentationPowerPoint.Show.89q %_% NChris HeadleeChris HeadleeOh+'0 X`p  Slide 1Chris HeadleeoPicturesCurrent UserGSummaryInformation((PowerPoint Document($ #&Chris Headleeo12iMicrosoft PowerPointP@NW@N@;՜.+,0    KOn-screen Show -s#&: Arial WingdingsDefault DesignSlide 1 ( `/ 0DArialngstt+0B 0"DWingdingstt+0B 0@ .  @n?" dd@  @@`` (  0AA @3ʚ;ʚ;g4\d\d B 0pTp<4dddd8pC 0t+ 80___PPT10   ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> f(    6x  `}  T Click to edit Master title style! !  04{  `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0 ^ `  >*  0 ^   @*  0l ^ `  @*H  0޽h ? 3380___PPT10.K tv Default Designh  h(   E <0O I DLimit - Definition  6tS  Definition: Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write if for every number  > 0 there is a number  > 0 such that |f(x)  L| <  whenever 0 < |x  a| <  Another way: if 0 < |x  a| <  then |f(x)  L| <  Alternatively: means that for every  > 0 (no matter how small) we can find  > 0 such that if x lies in the open interval (a - , a + ) and x `" a, then f(x) lies in the open interval (L  , L + )Lw Ni" "  &&"&"     $&$M((,",X> ",  M(4!D  6rKI  B lim f(x) = L xa:"   > 8 VT~ = ZB  s *Du/T/ZB  s *DpV  <,UK-V  3y  <QNK 3x   BCDEFAAjJ$m_@  7i,~TB  c $Dig i)TB  c $D  TB B c $D  TB B c $D ` n2  0"`- g t2  6"`+ e t2  6"` 'Gn2  0"`   <N0i 3a  <bN Hd p a - ,&  < KH p a + ,&  <N! a  p L - ,&  <vN }c  p L + ,&  <`N   _L$"D  6Ke u  B lim f(x) = L xa:"   >H  0޽h ? 3380___PPT10.K tvr0w e%DocumentSummaryInformation8"_1118936452FEUEUOle *CompObj+f Fonts UsedDesign Template Slide Titles FMicrosoft Equation 3.0 DS Equation Equation.39q\ЖII 'lim x!1 x 3 "1x"1ObjInfo-Equation Native .x_1118937104FEUEUOle 0 FMicrosoft Equation 3.0 DS Equation Equation.39qHII 'lim x!1 x 3 "1x"1='lim x!1 x"1()x 2 +x+1()x"1CompObj1fObjInfo3Equation Native 4d_1118938372FEUEU='lim x!1 x 2 +x+1()=3 FMicrosoft Equation 3.0 DS Equation Equation.39qTmIyI 'lim x!4 3x"7()=5Ole :CompObj;fObjInfo=Equation Native >p_1119013050FEUHUOle @CompObjAfObjInfoC FMicrosoft Equation 3.0 DS Equation Equation.39qg,II x"3<6!6x"3<!6x"18<!6x"5()"13<Equation Native D_1246965721dO)HULUOle HEPRINTD  dO)"Microsoft PowerPoint PresentationMSPresentationPowerPoint.Show.89q %_*  Chris HeadleeChris HeadleeOh+'0 X`p  ! & l1'% EMF\KhCF, EMF+@F\PEMF+"@ @ $@ 0@?!@ @ F(GDICFXLEMF+*@$?? @$yD;D!b $$=='% % V0.x>.x>% % $$AA" FEMF+@ FGDICF(GDIC1FGDICF(GDICUFGDICF(GDIC D:FGDICRp@"Arial04 \m0ǥd m> 0z08g 8t @{8 0@{ 0$l`p`0lTq(@{(dv%    T84KKKAJA. L`Continuity   % ( F(GDICIFGDICRp@"Arial0 <fh>0#0-o*ho0-z08gT0I0i0`-o@{dvdv`@"Arialw3 H3 dv%    T!PiKKKAJA!e L`Definition   % ( &% !h6h% ( Rp@"Arialo*h@| Q|Hm|3 d|0Q|Hm|30@{z08g` @"Arialw 3 dv!fPD|p|m|Ztm|dv%    T\PPiKKKAJAe-L: A function is continuous at a number a if              % ( Rp@"Arial0 nfh0 @{JTiK0W0fhȥ@{z08g佥K0@{!0l`pm50l r(@{,<dv%    T!KKKAJA!@LNote: that the definition implicitly requires three things of th               % ( Rp@"ArialW0fhL|Q|Hm| o0@{!0z08gm50l (@{,<dv!f\P|p|m|Z|m|dv%    TKKKAJA L`e function    % ( Rp@"Arial>0#0-o*ih0-0T0Dz08h`-o@{dvdvdvdv@"Arialw  p |0 dv%    TX!3KKKAJA!LP1.  % ( Rp@"Arial|-w!f`|-w 0w !f|6w;w 0w G-nwNnw 8Dhz08g 0D@{@"Arial@{m5000h8!赥Хdv%    T`GaKKKAJAGLTf(a  % ( Rp@"Arialw G-nwL|Q|Hm| 0D@{@"z080im5000 8!赥Хdv!f\P|p|m|Z|m|dv%    TTbKKKAJAb,L) is defined (i.e., a is in the domain of f)            % ( Rp@"Arial>0#0-o*0ih0-0T0Dz08h`-o@{dvdvdvdv@"Arialw  p |0 dv%    TX!3KKKAJA!LP2.  % ( Rp@"Arial>0#0-o*0ih0-0T0Dz08h`-o@{dvdvdvdv@"Arialw  p |0 dv%    TX!3KKKAJA!LP3.  % ( Rp@"Arial|/0[]@{m]\/0\z08g,m0m4\l`pm50lq:(@{,(:dv%    T!KdKKKAJA!`@Lf has a discontinuity at a, if f is not continuous at a. Note t                  % ( Rp@"Arial]L|Q|Hm| 0m4\z08gm50l :(@{,(:dv!f\P|p|m|Z|m|dv%    TKdKKKAJA`Lhe graphs of the examples of        % ( Rp@"Arial0 fg0 @{TiK0W0fg ȥ@{z08gн0@{!0l`pm50lq(@{(dv%    TGf0KKKAJAG{Lxdiscontinuities below:       % ( F(GDICsFwFGDICRp@"Arial0 fj0 @{TiK0W0fj$ȥ@{z08gH0@{!0l`pm50ls (@{ dv%    T`LiKKKAJAdLTlim % ( Rp@"ArialW0fjL|Q|Hm| 0@{!0z08hm50l (@{ dv!f\P|p|m|Z|m|dv%    T`LiKKKAJAdLTf(x  % ( Rp@"ArialL|Q|Hm| m50l (@{z080idv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    TL*iKKKAJAd Ld) =  % ( Rp@"Arial L|Q|Hm| !f \P|p|m|z08idv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    T`+LHiKKKAJA+dLTf(a % ( Rp@"Arial \L|Q|Hm| !f \P|p|m|z08Hjdv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    TTILPiKKKAJAIdLP) % ( Rp@"Arial0 f0i0 @{ TiK0W0f0i ȥ@{z08g 0@{!0l`pm50lr(@{|ddv%    TTkzKKKAJAxLPx % ( Rp@WingdingsW0f0iL|Q|Hm| 0@{!0z08hm50l (@{|ddv!f\P|p|m|Z|m|dv%    TTlzKKKAJAxLP % ( Rp@"ArialL|Q|Hm| m50l (@{|z08hdv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    TTkzKKKAJAxLPa % ( F(GDIC7:0FGDICRp@"Arial0 fj0 @{TiK0W0fjȥ@{z08gH0@{!0l`pm50ls(@{dv%    T`IgKKKAJAILTlim % ( Rp@"ArialW0fjL|Q|Hm| 0@{!0z08hm50l (@{dv!f\P|p|m|Z|m|dv%    T`nKKKAJAnLTf(x  % ( Rp@"ArialL|Q|Hm| m50l (@{z080idv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    TpKKKAJALX) =   % ( Rp@"Arial L|Q|Hm| !f \P|p|m|z08idv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    T`KKKAJALTf(a  % ( Rp@"Arial \L|Q|Hm| !f |p|m|z08Hjdv!f @Pp|7|{|{|@\ P|p|m|Z|m|dv%    TTKKKAJALP) % ( Rp@"Arial0 f0i0 @{ TiK0W0f0i ȥ@{z08g 0@{!0l`pm50lr(@{|ddv%    TT^e)KKKAJA^'LPx % ( Rp@WingdingsW0f0iL|Q|Hm| 0@{!0z08hm50l (@{|ddv!f\P|p|m|Z|m|dv%    TTeq)KKKAJAe'LP  % ( Rp@"ArialL|Q|Hm| m50l (@{|z08hdv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    TTrx)KKKAJAr'LPa % ( F(GDIC59FGDICRp@"Arial0 !fHj0 @{TiK0W0fHjȥ@{z08g40@{!0l`pm50l|s(@{́dv%    T`GeKKKAJAGLTlim % ( Rp@"ArialW0fHjL|Q|Hm| "0@{!0z08hm50l (@{́dv!f\P|p|m|Z|m|dv%    T`lKKKAJAlLTf(x  % ( Rp@"ArialL|Q|Hm| #m50l (@{́z080idv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    T`KKKAJALT)  % ( Rp@"Arial L|Q|Hm| $!f \P|p|m|z08idv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    TxKKKAJAL\exisits    % ( Rp@"Arial0 f0i0 @{ TiK0W0f0i ȥ@{z08g 0@{!0l`pm50lr(@{|ddv%    TT\cKKKAJA\LPx % ( Rp@WingdingsW0f0iL|Q|Hm| 0@{!0z08hm50l (@{|ddv!f\P|p|m|Z|m|dv%    TTcoKKKAJAcLP  % ( Rp@"ArialL|Q|Hm| m50l (@{|z08hdv!f \P|p|m|Z|m|dv!f\P|p|m|Z|m|dv%    TTpvKKKAJApLPa % ( F(GDIC'F(GDICKXF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, 9ΡBr'D1BC@( !b $$==_888% '%  ;)6"X4 #6)X4) )))))=2Y$@2=<?LV % % $$AA( " FEMF+@ FGDICF(GDIC'cpF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, BWDaCWD@!b $$==_888% %  ;u&6 &X4 & & & & & &6u&X4n&i&i&i&n&u&= `&Y$& &=<?%dn % % $$AA( " FEMF+@ FGDICFGDICF(GDIC&F(GDICKWF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, r$Cr'DV}CC@!b $$==_888% %  ;)6"X4#6)X4))))))=2Y$62=<?LU % % $$AA( " FEMF+@ FGDICF(GDIC&cpF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, ǑCWD0CWD@!b $$==_888% %  ;k&6&X4&&&&&&6k&X4e&`&`&`&e&k&=`&Y$&&=<?$dn % % $$AA( " FEMF+@ FGDICFGDICF(GDIC,F(GDICQ]F EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@ , Dr'DrDC@ !b $$==_888% %  ;T%)6_%"X4_%d%j%p%u%u%#6k%)X4k%)f%)_%)Y%)T%)T%)=<%2Y$k%%2=<?R\ % % $$AA( " FEMF+@ FGDICF(GDIC,cpF EMF+*@$??@ ^DZ?Ad8^@ A?( UU@UUUUUU@ , R DWD99DWD@  !b $$==_888% %  ;"&6.&X4.&.&.&.&.&.&6"&X4"&"&"&"&"&"&=.`&Y$a.&.&=<?*dn % % $$AA( " FEMF+@ FGDICFGDICF(GDIC.F(GDICR_F EMF+*@$??@ ^DZ?Ad8^@ A?( UU@UUUUUU@ , UDr'D9&VDC@  !b $$==_888% %  ;l5)6w5"X4w5|55555#65)X45)}5)w5)q5)l5)l5)=S52Y$552=<?S] % % $$AA( " FEMF+@ FGDICF(GDIC.cpF EMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUU@, rKDWDzDWD@!b $$==_888% %  ;2&6+>&X41>&6>&6>&6>&1>&+>&62&X42&2&2&2&2&2&=>`&Y$x>&>&=<?,dn % % $$AA( " FEMF+@ FGDICFGDICF(GDICQFGDICRp@"Arial0#\m0ǥd m> 0z08g8t@{8 0@{0$l`pm50lq(@{(dv%    T\KKKAJA\ L`Removable    % ( F(GDIC\FGDICRp@"Arial0#\m0ǥd m> 0z08g8t@{8 0@{0$l`pm50lq(@{(dv%    TgKKKAJAg L`Removable    % ( F(GDICYFGDICRp@"Arial0#\m0ǥdm> 0z08g8t@{8 0@{0$l`pm50lq(@{(dv%    T|dKKKAJAdL\Infinite    % ( F(GDICiFGDICRp@"Arial0#\m0ǥdm> 0z08g8t@{8 0@{0$l`pm50lq(@{(dv%    TdtKKKAJAtLTJump   % ( F(GDIC#wFtEMF+*@$??@^DZ?Ad8^@ A?( UU@UUUUUUd8^@ A?( UU@UUUUUU@, BKDFCTC@!b $$==_888% %  ;&6! 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T`3KKKAJALTlim % ( Rp@"Arial4L|XQ|Hm|o ``8t3@{8 0@{30PU0z08hm50l ( @{z dv r !G\P|p|m|Z|m|dv%    T`9PKKKAJA9LTf(x  % ( Rp@"Arial3@{8L|XQ|Hm|o ``m50l ( @{0PU0z08hdvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    TdQlKKKAJAQLT) =   % ( Rp@"Arial L|XQ|Hm|o ``r !G \P|p|0PU0z080idvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    T`mKKKAJAmLTlim % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08Hjdvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    TKKKAJALt-------------------- % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08jdvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    TX)KKKAJALP=  % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08`kdvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    T`*FKKKAJA*LTlim % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08ߥ dvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    TlLkKKKAJALLX----- % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08x dvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    T`qKKKAJAqLT= 2   % ( Rp@"Arial0i fj0 D@{7TiK0W0fjNzD0PU0z08g A 80t@ @{!0l`pm50ls(@{dv%    TTKKKAJALPx  % ( Rp@"ArialW0fjL|XQ|Hm|i ``0t@ @{0PU0z08gm50l (@{dvr !G\P|p|m|Z|m|dv%    TTKKKAJALP % ( Rp@"Arialt@ L|XQ|Hm|i ``m50l (@{0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TXKKKAJALP(5  % ( Rp@"Arial L|XQ|Hm|i ``r !G \P|p|0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TdKKKAJALT+ 1/   % ( Rp@"Arial \L|XQ|Hm|i ``r !G \P|p|0PU0z080idvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTKKKAJALPx  % ( Rp@"Arial \L|XQ|Hm|i ``r !G \P|p|0PU0z08idvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTKKKAJALP % ( Rp@"Arial \L|XQ|Hm|i ``r !G \P|p|0PU0z08Hjdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TcKKKAJALt) 5  % ( F(GDICMFGDICRp@"Arial0\m0z`m0PU0z08g8t@{8 0@{0$l`pm50l r(@{,<dv%    TT(.KKKAJA)LPx % ( Rp@"ArialL|XQ|Hm|o ``8t@{8 0@{0PU0z08gm50l (@{,<dvr !G\P|p|m|Z|m|dv%    TX/AKKKAJA/LP!"  % ( F(GDIChFGDICRp@"Arial0\m0z`m0PU0z08g8t@{8 0@{0$l`pm50l r(@{,<dv%    TTrxKKKAJAsLPx % ( Rp@"ArialL|XQ|Hm|i ``8t@{8 0@{0PU0z08gm50l (@{,<dvr !G\P|p|m|Z|m|dv%    TXyKKKAJAyLP!"  % ( F(GDIC'WFGDICRp@"Arial0\m0z`m0PU0z08g8t@{8 0@{0$l`pm50l r(@{,<dv%    TT17KKKAJA2LPx % ( Rp@"ArialL|XQ|Hm|o ``8t@{8 0@{0PU0z08gm50l (@{,<dvr !G\P|p|m|Z|m|dv%    TX8JKKKAJA8LP!"  % ( F(GDIC2{FGDICRp@"Arial0o <fj>0#0-xz*j0PU0z08gT0I0i0 -xz@{dvdv @"Arialwi *Hi *dv%    TT|KKKAJALPx  % ( Rp@"Arialz*j@|X Q|Hm|i +``I0i0 -xz0PU0z08g  @"Arialwi *i *dvr !GPD|p|m|Z|m|dv%    TT{KKKAJALP4 % ( Rp@"Ariali0 -@|X Q|Hm|i ,``  0PU0z08hdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    T`|KKKAJALT(16  % ( Rp@"Arial @|X Q|Hm|i -``r !G PD|p|0PU0z08hdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    Td|KKKAJALT+ 1/   % ( Rp@"Arial P@|X Q|Hm|i .``r !G PD|p|0PU0z080idvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    TT|!KKKAJALPx  % ( Rp@"Arial P@|X Q|Hm|i /``r !G PD|p|0PU0z08idvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    TT"|'KKKAJA"LP % ( Rp@"Arial P@|X Q|Hm|i 0``r !G PD|p|0PU0z08Hjdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    T(|KKKAJA(Lp) 16   % ( Rp@"Arial0# 3\m 0z`4m0PU0z08g38t3@{8 0@{30$l`p m50l z( @{z dv%    T`=XKKKAJA=LTlim % ( Rp@"Arial4L|XQ|Hm|o 4``8t3@{8 0@{30PU0z08hm50l ( @{z dv r !G\P|p|m|Z|m|dv%    T`^zKKKAJA^LTg(x   % ( Rp@"Arial3@{8L|XQ|Hm|o 5``m50l ( @{0PU0z08hdvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    Td{KKKAJA{LT) =   % ( Rp@"Arial L|XQ|Hm|o 6``r !G \P|p|0PU0z080idvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    T`KKKAJALTlim % ( Rp@"Arial \L|XQ|Hm|o 7``r !G \P|p|0PU0z08Hjdvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    T8KKKAJALt-------------------- % ( Rp@"Arial \L|XQ|Hm|o 8``r !G \P|p|0PU0z08jdvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    TXDTKKKAJADLP=  % ( Rp@"Arial \L|XQ|Hm|o 9``r !G \P|p|0PU0z08`kdvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    T`UpKKKAJAULTlim % ( Rp@"Arial \L|XQ|Hm|o :``r !G \P|p|0PU0z08ߥ dvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    TlvKKKAJAvLX----- % ( Rp@"Arial \L|XQ|Hm|o ;``r !G \P|p|0PU0z08x dvr !G \P|p|m|Z|m|dv r !G\P|p|m|Z|m|dv%    T`KKKAJALT= 4   % ( Rp@"Arial0o K<fj>0#0-xz*j0PU0z08gT0I0i0 -xz@{dvdv @"Arialwi WHi Wdv%    TTKKKAJALPx  % ( Rp@"Arialz*j@|X Q|Hm|i X``I0i0 -xz0PU0z08g  @"Arialwi Wi Wdvr !GPD|p|m|Z|m|dv%    TTKKKAJALP4 % ( Rp@"Ariali0 -@|X Q|Hm|i Y``  0PU0z08hdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    TXKKKAJALP(4  % ( Rp@"Arial @|X Q|Hm|i Z``r !G PD|p|0PU0z08hdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    Td KKKAJALT+ 7/   % ( Rp@"Arial P@|X Q|Hm|i [``r !G PD|p|0PU0z080idvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    TT KKKAJA LPx  % ( Rp@"Arial P@|X Q|Hm|i \``r !G PD|p|0PU0z08idvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    TTKKKAJALP4 % ( Rp@"Arial P@|X Q|Hm|i ]``r !G PD|p|0PU0z08Hjdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    TKKKAJALt) 4  % ( F(GDICBuFGDICRp@"Arial0\m0z`m0PU0z08g8t@{8 0@{0$l`pm50lr(@{|ddv%    TTLRKKKAJAMLPx % ( Rp@"ArialL|XQ|Hm|o U``8t@{8 0@{0PU0z08gm50l (@{|ddvr !G\P|p|m|Z|m|dv%    TTS]KKKAJASLP!  % ( Rp@"Arial@{8L|XQ|Hm|o V``m50l (@{0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TT^aKKKAJA^LP- % ( Rp@"Arial L|XQ|Hm|o W``r !G \P|p|0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTbiKKKAJAbLP" % ( F(GDICFGDICRp@"Arial0\m0z`m0PU0z08g8t@{8 0@{0$l`pm50lr(@{|ddv%    TTKKKAJALPx % ( Rp@"ArialL|XQ|Hm|i q``8t@{8 0@{0PU0z08gm50l (@{|ddvr !G\P|p|m|Z|m|dv%    TTKKKAJALP!  % ( Rp@"Arial@{8L|XQ|Hm|i r``m50l (@{0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTKKKAJALP- % ( Rp@"Arial L|XQ|Hm|i s``r !G \P|p|0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTKKKAJALP" % ( F(GDICLFGDICRp@"Arial0\m0z`m0PU0z08g8t@{8 0@{0$l`pm50lr(@{|ddv%    TTV\KKKAJAWLPx % ( Rp@"ArialL|XQ|Hm|o k``8t@{8 0@{0PU0z08gm50l (@{|ddvr !G\P|p|m|Z|m|dv%    TT]gKKKAJA]LP!  % ( Rp@"Arial@{8L|XQ|Hm|o l``m50l (@{0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TThkKKKAJAhLP- % ( Rp@"Arial L|XQ|Hm|o m``r !G \P|p|0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTlsKKKAJAlLP" % ( F(GDIC"oFGDICRp@"Arial0o w<f0i>0#0-z*0i0PU0z08gT0I0i0 -z@{dvdv0lr\t(0dv%    T`gqKKKAJAgLT16x  % ( Rp@"Arialz*0i@|X Q|Hm|i ``I0i0 -z0PU0z08g0l \t(0dvr !GPD|p|m|Z|m|dv%    TTpKKKAJA}LP4 % ( Rp@"Ariali0 -@|X Q|Hm|i ``0l \t0PU0z08hdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    T`qKKKAJALT+ x   % ( Rp@"Arial @|X Q|Hm|i ``r !G PD|p|0PU0z08hdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    TTqKKKAJALP % ( Rp@"Arial0#\m0z`m0PU0z08g8t@{8 0@{0$l`pm50lhr(@{TPdv%    T`-IKKKAJA-LTg(x   % ( Rp@"ArialL|XQ|Hm|o ``8t@{8 0@{0PU0z08gm50l (@{TPdvr !G\P|p|m|Z|m|dv%    TdJeKKKAJAJLT) =   % ( Rp@"Arial@{8L|XQ|Hm|o ``m50l (@{0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TfKKKAJAf Lh------------- % ( Rp@"Arial0o <fh>0#0-z*h0PU0z08gT0I0i0 -z@{dvdv @"Arialwi Hi dv%    TXrKKKAJArLP4x  % ( Rp@"Arialz*h@|X Q|Hm|i ``I0i0 -z0PU0z08g  @"Arialwi i dvr !GPD|p|m|Z|m|dv%    TTKKKAJALP4 % ( Rp@"Ariali0 -@|X Q|Hm|i ``  0PU0z08hdvr !G PD|p|m|Z|m|dvr !GPD|p|m|Z|m|dv%    T`KKKAJALT+ 7   % ( F(GDIC59<@FEMF+*@$??@ NDۜC!MDۜCxMDkCxMDCxMDϞC!MD`CND`CrND`C9NDϞC9NDC9NDkCrNDۜCNDۜC@!b $$==%  ;UP 3i3V3V3V3i33333333<>59;? % $$AAF`TEMF+@@4^UU?A@$$==_888% % UP37<@ 3i3V3V3V3i33333333% % $$AA( " FEMF+@ FGDICF(GDIC(FEMF+*@$??@L@^UU?AA@@@, 9^ DXzCbDXzC@!b $$==_888% %  ;"6#X4 # # # # ##6"X4""""""=G#6#X4######6G#X4C#?#?#?#C#G#=#6s$X4x$|$|$|$x$s$6#X4######=$6+%X4/%3%3%3%/%+%6$X4$$$$$$=m%6%X4%%%%%%6m%X4i%e%e%e%i%m%=%&6&X4&&&&&&6%&X4 &&&& &%&=&6Q'X4U'Y'Y'Y'U'Q'6&X4&&&&&&='6(X4 (((( ((6'X4''''''=K(6(X4((((((6K(X4F(B(B(B(F(K(=)6w)X4{)))){)w)6)X4((((()=)6.*X43*6*6*6*3*.*6)X4))))))=q*6*X4******6q*X4l*h*h*h*l*q*=(+6+X4++++++6(+X4#+ + + +#+(+=+6T,X4Y,\,\,\,Y,T,6+X4++++++=,6 -X4----- -6,X4,,,,,,=N-6-X4------6N-X4I-F-F-F-I-N-=.6z.X4.....z.6.X4.---..=.61/X46/:/:/:/6/1/6.X4......=t/6/X4//////6t/X4o/l/l/l/o/t/=+060X40000006+0X4'0#0#0#0'0+0=06W1X4\1`1`1`1\1W160X4000000=162X422222261X4111111=Q262X42222226Q2X4M2I2I2I2M2Q2= 36}3X433333}36 3X433333 3=3654X494=4=4=4945463X4333333=w464X44444446w4X4s4o4o4o4s4w4=/565X45555556/5X4*5&5&5&5*5/5=56[6X4_6c6c6c6_6[665X4555555=667X477777766X4666666=U767X47777776U7X4P7L7L7L7P7U7= 868X48888886 8X488888 8=<?' % % $$AA( " FEMF+@ FGDICF(GDIC7+BF(GDIC7BFEMF+*@$??@@4^UU?A@, 9DЛC9D9>C@!b $$==_888% % W$5Bi.si.% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@@4^UU?A@, U 3DЛCU 3D9>C@!b $$==_888% % W$5B,s,% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@@4^UU?A@, DZ/DЛCDZ/D9>C@!b $$==_888% % W$5B+s+% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@@4^UU?A@, O,DЛCO,D9>C@!b $$==_888% % W$5B +s +% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@@4^UU?A@, (DЛC(D9>C@!b $$==_888% % W$5B6*s6*% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@@4^UU?A@, %DЛC%D9>C@!b $$==_888% % W$5B^)s^)% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@@4^UU?A@, l6DЛCl6D9>C@!b $$==_888% % W$5B-s-% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@ @4^UU?A@!, 8GDЛC8GD9>C@! !b $$==_888% % W$5B1s1% % $$AA( " FEMF+@ FGDICF(GDIC)7+BFEMF+*@$??@"@4^UU?A@#, JDЛCJD9>C@#"!b $$==_888% % W$(5,B2s2% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@$@4^UU?A@%, rCDЛCrCD9>C@%$!b $$==_888% % W$ 5B0s0% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@&@4^UU?A@', }@DЛC}@D9>C@'&!b $$==_888% % W$5B0s0% % $$AA( " FEMF+@ FGDICF(GDIC7BFEMF+*@$??@(@4^UU?A@), 9&=DЛC9&=D9>C@)(!b $$==_888% % W$5BB/sB/% % $$AA( " FEMF+@ FGDICFGDICF(GDIC&!-(FEMF+*@$??@* UGJDʐCZIDʐC9IDZC9ID C9IDܾCZIDOCUGJDOCQJDOCrJDܾCrJD CrJDZCQJDʐCUGJDʐC@*!b $$==%  ;UP 2t2b2$b2:b2Pt2b2b2b2P2:2$22<>&!,' % $$AAF`TEMF+@+@4^UU?A@*+$$==_888% % UP$-( 2t2b2$b2:b2Pt2b2b2b2P2:2$22% % $$AA( " FEMF+@ FGDICF(GDIC!FEMF+*@$??@, ^GDCGDCǽFDKCǽFDCǽFDCGD@C^GD@C߷GD@CHDCHDCHDKC߷GDC^GDC@,!b $$==%  ;UP 111"181N1`1`1`1N181"11<>  % $$AAF`TEMF+@-@4^UU?A@,-$$==_888% % UP! 111"181N1`1`1`1N181"11% % $$AA( " FEMF+@ FGDICF(GDIC FEMF+*@$??@/ CD7{C̮BD7{CfBDW|CfBD}CfBD}C̮BD CCD C`CD CCD}CCD}CCDW|C`CD7{CCD7{C@/!b $$==%  ;UP 0000000000000<>  % $$AAF`TEMF+@.@4^UU?A@/.$$==_888% % UP 0000000000000% % $$AA( " FEMF+@ FGDICF(GDIC*9=FXLEMF+*@$??@04(UU@A@1UMDНCMDۜCVMDrCLDCLDrC9bLDC*LD CrLKDUC9nJD{CIDCcID;CIDrTCBHDCrdGDCVEDV5CVBD~C9>9D8wC+D9n|CV="DV|CxD!yC2!D{C D{C@10!b $$==_888!% % Ut):>x3Y3N3m53N'3;30322hu2 Q2x2 21910H.l*(W&E("% % $$AA( " FEMF+@ FGDICF(GDIC FGDICRp@"Arial0#\m0z`m0PU0z08g8t@{8 0@{0$l`pm50lq(@{(dv%    TlKKKAJALXy = 4    % ( F(GDICDFGDICRp@"Arial0#!\m0z`"m0PU0z08g!8t!@{8 0@{!0$l`pm50lq(@{(dv%    TOKKKAJAO"LRemember infinity is not a number!            % ( F(GDICFGDICRp@"Arial0#2\m0z`3m0PU0z08g28t2@{8 0@{20$l`pm50lě (@{ dv%    T`)DKKKAJA)LTlim % ( Rp@"Arial3L|XQ|Hm|o ``8t2@{8 0@{20PU0z08gm50l (@{ dvr !G\P|p|m|Z|m|dv%    TXJZKKKAJAJLP(x  % ( Rp@"Arial2@{8L|XQ|Hm|o ``m50l (@{0PU0z08hdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TT[aKKKAJA[LP % ( Rp@"Arial L|XQ|Hm|o ``r !G \P|p|0PU0z080idvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTglKKKAJAgLP- % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08idvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    T`rKKKAJArLT5x)  % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08Hjdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TXKKKAJALP=  % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08jdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    T`KKKAJALTlim % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08`kdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTKKKAJALPx  % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08kdvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTKKKAJALP % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08x dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTKKKAJALP- % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08 dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TXKKKAJALP5  % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08 dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    T`KKKAJALTlim % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08( dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TT$KKKAJALPx  % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08 dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TX*:KKKAJA*LP=  % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08H dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TT;HKKKAJA;LP" % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08 dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    T`YvKKKAJAYLTnot  % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08 dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TT|KKKAJA|LP" % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z08 dvr !G \P|p|m|Z|m|dvr !G\P|p|m|Z|m|dv%    TTKKKAJALP- % ( Rp@"Arial \L|XQ|Hm|o ``r !G \P|p|0PU0z088 dvr !G 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X4      = 6 X4      6 X4      =B }6 iX4 g j p v | ~6I X4C = ; 9 = B }= N6:X48";$A&G#MO6 cX4 e b \ V P N=k6 X4   6r4X4l6f3d-b'e!k= 6? X4D K M N K E 6X4    = 6 X4      6 X4      =( 6g} X4m{ s~ u w t n 6. X4( "   " ( =b 6N X4L O U [ a c 6w X4y v p j d b =P3 6 X4  & , 2 4 6WH X4QJ KG IA G; J5 P3 = 6$ X4* 0 2 4 0 + 6 X4      =x 6 X4      6 X4y s q o s x =  6L X4R X Z \ Y S 6 X4      =v 6b X4` c i o u w 6 X4   ~ x v =5G 6u3 X4{1 4 : @ F {H 6<\ X46^ 0[ .U ,O /I 5G = 6  X4      6- X4/ , &    =^ 6 X4      6d X4_ X V U X ^ = 61 X47 = ? A > 8 6 X4      = 6v X4t x }    6 X4   }   =[ 6ZG X4`E fH hN jT fZ a\ 6!p X4r o i c ] [ =, 6 X4   % + - 6A X4C @ : 4 . , =C 6 X4      6J X4D = < : = C = 6 X4 # % & #  6 X4      =k 6 X4      6r X4l f d b e k =o 6?[ X4EY K\ Mb Oh Ln Fp 6 X4  } w q o =@ 6, X4* - 3 9 ? A 6U X4W T N H B @ =( 6g X4m t u w t n 6/& X4)( #% !  " ( = 6 X4      6 X4      =P 6 X4      6W X4Q K I G K P = 6$o X4*m 0q 2v 4| 1 + 6 X4      =sY$Mc=V; Y$K r =<?C % % $$AA( " FEMF+@ FGDICF(GDICaFEMF+*@$??@ \P^UU?AA@@?@@?@@@ , C,Cr|B,C@  !b $$==_888% %  ; 6 X4 { { {   6 X4      =A 6A X4= 9 9 9 = A 6A X4F I I I F A = 6 X4      6 X4      = 6G X4B ? ? ? B G 6 X4      = 6 X4      6 X4       = 6 X4      6 X4      = 6  X4      6 X4      = 6 X4      6 X4      = 6 X4 } } }   6 X4      =B 6 X4      6B X4G K K K G B = 6 X4      6 X4      =H 6H X4D @ @ @ D H 6H X4M Q Q Q M H = 6 X4      6 X4       =N 6N X4J F F F J N 6N X4S W W W S N =  6  X4      6  X4      = 6T X4P L L L P T 6 X4      = 6 X4       6 X4      = 6 X4      6 X4      = 6 X4      6 X4      = 6 X4      6 X4      = 6 X4      6 X4      =P 6 X4      6P X4T X X X T P = 6 X4      6 X4      =V 6V X4Q M M M Q V 6V X4Z ^ ^ ^ Z V = 6 X4      6 X4      =\ 6\ X4W S S S W \ 6\ X4` d d d ` \ = 6 X4      6 X4 ! ! !   = 6a X4] Y Y Y ] a 6 X4 = 6 X4      6 X4# ' ' ' #  = 6 X4 6 X4 = 6% X4    % 6 X4 = 6 X4 6 X4 = 6 X4 6 X4 =] 6 X4 6] X4a e e e a ] = 6 X4 6 X4 =c 6c X4^ Z Z Z ^ c 6c X4g k k k g c = 6 X4 6 X4% ( ( ( % =i 6i X4d ` ` ` d i 6i X4m q q q m i =& 6& X4!    ! & 6& X4+ . . . + & = 6o X4j f f f j o 6 X4      =, 6, X4' $ $ $ ' , 6, X41 4 4 4 1 , = 6 X4      6 X4      = 62 X4- * * * - 2 6 X4      = 6 X4      6 X4      = 6 X4      6 X4      =j 6 X4      6j X4o r r r o j =<?_ % % $$AA( " FEMF+@ FGDICF(GDICbFEMF+*@$??@\P^UU?AA@@?@@?@@@, #CC9^nC8B9^nC@!b $$==_888% %  ;+ 6 X4      6+ X4/ 3 3 3 / + =s 6s X4o k k k o s 6s X4x | | | x s =1 61 X4, ( ( ( , 1 61 X45 9 9 9 5 1 = 6y X4u q q q u y 6 X4      =7 67 X42 . . . 2 7 67 X4; ? 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S  0|} ^ ` } >*  0H} ^  } @*  0 } ^ ` } @*H  0޽h ? 3380___PPT10.K tv Default Design0 ,$  (   E <}Ok I @Differentiable  6}  Definition: A function is differentiable at a if f (a) exists. It is differentiable on an open interval (a, b) if it is differentiable at every number in the interval. Thrm: If f is differentiable at a, then f is continuous at a. Note: The converse is not necessarily true a function can be continuous at a, but not be differentiable at a f has a discontinuity at a, if f is not continuous at a. 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