ࡱ> uY .bjbj .xx#  cc$$$$$$$ %4$ ( ///c/<<<%''''''$@K$h<Kcc/c/u`4c /L$c/%%:,ic/dh$C, v0Roi$i<Z@֧4 <<<KKg&<<<<<<<<<<<<  ): AP Calculus Notes: Limits 1 Ex:  EMBED Equation.DSMT4  a. Natural domain: b. What happens at x = 1?  c. What happens as x ( 1? Note: ( is read approaches or goes to. x ( 1 means x gets arbitrarily close to 1, from either side, but never exactly equals 1. 1. Graph it. (Graph at right.) 2. Make a table.  EMBED Equation.DSMT4   Notation: x(1 : x(1+ : Notation: Limit from the left: Limit from the right: Important notes: 1. This does not mean that f(1) = 2; f(1) is still undefined. 2. While the graph and table are both very convincing, they do not prove that the limit is 2. Ideas of a Limit To generalize:  EMBED Equation.DSMT4  means 1. Informally: As x approaches a, f(x) approaches L: As x ( a, f(x) ( L. Note: In the actual definition of a limit, x can never exactly equal a. It is possible however, that f(x) might exactly equal L in some limits. 2. Better: By getting x close enough to a (but not equal to a), we can make y as close as we like to L (but not necessarily equal to L). Ex: Use the graphs to evaluate the following limits:  a.  EMBED Equation.DSMT4   b.  EMBED Equation.DSMT4   c.  EMBED Equation.DSMT4   d.  EMBED Equation.DSMT4  Ex: A function f is shown in the graph at right. The domain of the function is all real numbers except x = 4 and x = 0. Use the graph to evaluate the following:  a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4  i.  EMBED Equation.DSMT4  j.  EMBED Equation.DSMT4  k.  EMBED Equation.DSMT4  AP Calculus Notes: Limits 2 Properties of Limits Suppose k is a constant and  EMBED Equation.DSMT4  = L and  EMBED Equation.DSMT4  = M . Then the following properties hold: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  7. If f is continuous at M, then  EMBED Equation.DSMT4  Ex: If  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , evaluate a.  EMBED Equation.DSMT4 = b.  EMBED Equation.DSMT4 = c.  EMBED Equation.DSMT4 = d.  EMBED Equation.DSMT4 = e.  EMBED Equation.DSMT4 = f.  EMBED Equation.DSMT4 = g.  EMBED Equation.DSMT4 = h.  EMBED Equation.DSMT4 = Evaluating Limits by Direct Substitution Ex: Use the properties of limits to evaluate  EMBED Equation.DSMT4   EMBED Equation.DSMT4 = Many (not all!) limits can be evaluated by direct substitution. Ex:  EMBED Equation.DSMT4 = Ex:  EMBED Equation.DSMT4 = Ex:  EMBED Equation.DSMT4 =  When doesnt it work? Ex:  EMBED Equation.DSMT4   Ex:  EMBED Equation.DSMT4 where [x] is the greatest integer function: [x] = the greatest integer ( x.  Ex:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 where  EMBED Equation.DSMT4  AP Calculus Notes: Limits 3 Evaluating Limits Algebraically If a function f is continuous at x = a (there is no break in the graph of f at x = a), then the limit  EMBED Equation.DSMT4 can be evaluated by direct substitution:  EMBED Equation.DSMT4 . Direct substitution does not work if f(a) evaluates to  EMBED Equation.DSMT4 . It does not necessarily mean the limit DNE. It does mean we need to do extra analysis (work). Our strategy will be to try to show that f is identical, except at x = a, to some other function g for which direct substitution does work. Ex: a)  EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4  d)  EMBED Equation.DSMT4  e)  EMBED Equation.DSMT4  f)  EMBED Equation.DSMT4  g)  EMBED Equation.DSMT4  Summary To evaluate a limit: 1. Try direct substitution. Unfortunately, this does not work for most of the important limits in calc. 2. If the function is piecewise defined with a break at the limit, evaluate the limit on both sides. See if the one-sided limits are the same. (See last example from Limits 2). 3. If direct substitution gives  EMBED Equation.DSMT4  where N (0, then the limit DNE (cant be fixed) (ex e above) 4. If direct substitution gives  EMBED Equation.DSMT4  then try appropriate algebraic tricks: a. factoring and reducing (ex a, e above) b. rationalizing numerator or denominator (ex c, d) c. simplifying complex fractions (ex b) d. multiplying out and simplifying (ex g) 5. If all else fails, look at graph and/or table. (Remember, these are good for exploring or checking but are not considered proofs.) AP Calculus Notes: Limits 5 Continuity Informal definition: A function is continuous at a point x = c if the graph of the function does not have a break there. A function is continuous on an interval (a, b) if it is continuous at each point in the interval; in other words, if it has no breaks in the interval.  Ex: a)  EMBED Equation.DSMT4   b)  EMBED Equation.DSMT4   c)  EMBED Equation.DSMT4   d)  EMBED Equation.DSMT4   e)  EMBED Equation.DSMT4  Continuity: Formal Definition A function f is continuous at the point x = c if 1. f(c) is defined 2.  EMBED Equation.DSMT4  exists 3.  EMBED Equation.DSMT4   Ex: b)  EMBED Equation.DSMT4   c)  EMBED Equation.DSMT4   e)  EMBED Equation.DSMT4  Ex: Find the value of a that will make f continuous at x = 2 if  EMBED Equation.DSMT4 . AP Calculus Notes: Limits 6 Intermediate Value Theorem Ex: You go for a hike in the Adirondack Mountains. At 8:00 AM you are at 1500 elevation; at noon you are at 3500 elevation. Between 8:00 AM and noon, can you say for sure if you were ever at an elevation of a. 2500'? b. 4000'? Intermediate Value Theorem (IVT): If f(x) is continuous on [a, b] and N is any value strictly between f(a) and f(b), then there is at least one value c in (a, b) such that f(c) = N.  In the Adirondack hiking example above, [a, b] = f(a) = f(b) = N = What (reasonable) assumption did we make about our elevation as a function of time? Ex: Does  EMBED Equation.DSMT4  have a root in [1, 2]? Ex: A continuous function g has domain 0 ( x ( 10. Certain values of g are shown in the table below.  What is the least possible number of solutions to the equation g(x) = 7 if a. k = 6 b. k = 7 c. k = 8 Ex: Does  EMBED Equation.DSMT4  = 1 somewhere in the interval [1, 3]? AP Calculus Notes: Limits 7 Infinite Limits Ex: Describe the behavior of  EMBED Equation.DSMT4  near x = 3. Notes: 1.  EMBED Equation.DSMT4  DNE. Saying it equals ( is simply telling more precisely why it DNE 2.  EMBED Equation.DSMT4  means As x approaches c, f(x) gets larger and larger without bound (with no limit). Ex: Ex: Describe the behavior of  EMBED Equation.DSMT4  near x = 3. Vertical Asymptotes If  EMBED Equation.DSMT4  or ( EMBED Equation.DSMT4 , or if either of the one-sided limits at x = c equals  EMBED Equation.DSMT4  or ( EMBED Equation.DSMT4 , then f has a vertical asymptote at x = c. Ex:  EMBED Equation.DSMT4  Ex:  EMBED Equation.DSMT4  Ex:  EMBED Equation.DSMT4  AP Calculus Notes: Limits 8 Limits at Infinity Ex: Evaluate  EMBED Equation.DSMT4 . The following facts are helpful when evaluating limits at ((: 1. For large x, a polynomial function behaves like its highest order term. 2. An exponential function ax (a > 1) grows faster than any power of x. 3. Any positive power of x grows faster than a log function logax (a > 1). Ex: Evaluate the following a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4  e.  EMBED Equation.DSMT4  f.  EMBED Equation.DSMT4  g.  EMBED Equation.DSMT4  h.  EMBED Equation.DSMT4  i.  EMBED Equation.DSMT4  Horizontal Asymptotes If  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 , then f has a horizontal asymptote at y = b. Ex: a.  EMBED Equation.DSMT4   b.  EMBED Equation.DSMT4   c.  EMBED Equation.DSMT4   d.  EMBED Equation.DSMT4  y y x y y x y x y x y x y x y x y x (  EMBED Equation.DSMT4  x y f(a) a f(b) b f(x)  EMBED Equation.DSMT4   EMBED Equation.DSMT4  x y y x y x y x x  x x  x0.90.990.9990.9999x ( 11.00011.0011.011.1y (1 1 x0246810g(x)11935k4 20 10 0 -10 -20 y  y x  EMBED Equation.DSMT4   6 5 4 3 2 1 -1 -2 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 x 1 -1 1 2 x x  x  x  %&=>?@kloprs      / A D E \ ] t u v w x ڒjh=0hH2iEHUh($h73whH2ih676] jh67jh+VUmHnHu h676]jh=0hH2iEHUj-\M hH2iCJUVhH2ijhH2iUh67hh67>*5 ABWXrt ! B C D E F Y Z x z { | } ~  gd67 ^gd67gd67gd67x y      $ % ( ) * + 6 @ ÿøøøøøôjh 1Uh 1hz 0h 15>*\h 15>*\hr5>*\hJhr h73w6]hz 0h73w h67H*h($ h4#h67 h#0H*hH2i jh67 h676h67jh67CJUmHnHu3 A B  gd 1 gdJ gd67         ! 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