ࡱ>  ybq` mbjbjqPqP R::V`7ZVVVV222FFFF8FF"HF52IW(WWZbbb$ghϋ\ڇ2Jc!abJcJcڇVVWZ!nnnJcVVW2ZnJcnn2|ZH ٛF d0405F6+g:+,|+2|bbnb bSbbbڇڇ$njbbb5JcJcJcJcFFF$<jDFFFjDFFFVVVVVV NOTES: LIMITS The limit is a notion of motion. It is the idea of approaching a specific x-value with the intent of seeing what the y-values are doing there. With the limit, we are not really specifically interested the ACTUAL y-value AT the specific x-value, but rather the INTENDED y-value. Imagine a bridge that spans a giant gorge. Now imagine that the bridge has been blown up and gone. When driving on the road from either side of the gorge, you can clearly see where the bridge WAS and where the gorge IS. You dont need to jump out into the middle of the gorge and risk injury just to see whats happening there. Now imagine that the bridge is intact. Once again, you dont need to set foot on the bridge to see where the gorge is. Therefore, the existence or non-existence of the bridge is irrelevant to the existence of the gorge. This is exactly why the limit is so important. It gives us a way to talk about the activity at a point whether or not the graph exists there or not. Heres an example: Lets say a road is defined by the graph of  EMBED Equation.DSMT4 . We want to know what the y-values are approaching as the x-values are approaching  EMBED Equation.DSMT4 , the x-coordinate of the gorge. From the left, it appears the y-values are approaching 4.  From the right, it also appears the y-values are approaching 4.  It would appear that the gorge occurs along the road at the coordinate  EMBED Equation.DSMT4  since the graph (the road) approach this point from either side it. Mathematically, we say:  EMBED Equation.DSMT4  If the graph approaches the same value from both sides (and ONLY if), we say the limit exists. In this case:  EMBED Equation.DSMT4  So whats actually happening AT this point  EMBED Equation.DSMT4 ? Is a there a bridge or no bridge? To determine this, we must see if the function value actually exists there and if it, in fact, equal to the limit value. In this example,  EMBED Equation.DSMT4   In this example, since  EMBED Equation.DSMT4 , there IS a bridge, and we can safely and easily cross there. Consequently, mathematically speaking, we say the graph is continuous at this point. Heres a summary: Theorem: Definition: When a function is continuous at a point, the function value and the limit value are the same. Knowing that, we can evaluate the limit simply by direct substitution (try finding  EMBED Equation.DSMT4  this way), but is this ALWAYS the case? Heres another example that explores this question. Find  EMBED Equation.DSMT4 . When we try direct substitution, we get the indeterminate form  EMBED Equation.DSMT4 , which means that the bridge is out, but we are still interested in finding y-coordinate of where it might have been. Here, all we need to do is factor the numerator, then divide out the bad guy factor of  EMBED Equation.DSMT4  who was causing all that unhappiness in the denominator. Once hes gone, we can use direct substitution.  EMBED Equation.DSMT4  *Notice that we must rewrite the limit notation up until we actually plug in the desired value. So why does this work? Great question! Its because the two functions are equivalent except at a single point, the value  EMBED Equation.DSMT4 . The rational function has a removable discontinuity there (a bridge thats out), while the function  EMBED Equation.DSMT4  has a function value there (a bridge). We can use the line, and its function value there, to find the limit value for the rational function. The algebraic method above is pretty nifty for doing this. Time to move on. Lets look at messy graph of  EMBED Equation.DSMT4 with many discontinuities. Call it graphus interruptus.  Discuss the continuity at the following values of x. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  11.  EMBED Equation.DSMT4   Find the following: 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  We can discuss continuity on an open interval and a closed interval, as well as at a point.  Looking at the graph again, decide if the following claims about continuity on the given intervals are true or false. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3.  EMBED Equation.DSMT4  4.  EMBED Equation.DSMT4  Now lets look at a piece-wise function. They are always interesting. Ex) For  EMBED Equation.DSMT4 , find the following: a)  EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4  d)  EMBED Equation.DSMT4  Notice the two pieces making up the function are polynomials, which are continuous everywhere. That means anywhere we wish to evaluate the limit, we may do so by direct substitution. Here we just need to be sure to plug into the correct piece. We can plug -1 into the top piece to get EMBED Equation.DSMT4  We plug -1 into the bottom piece to get  EMBED Equation.DSMT4  Since the limit from each side of  EMBED Equation.DSMT4  are the same, we say the limit exists and is 3, or mathematically,  EMBED Equation.DSMT4 . Here, we plug into the top piece to get  EMBED Equation.DSMT4  Since all of these conditions are met, and since  EMBED Equation.DSMT4 , we know the function is continuous at  EMBED Equation.DSMT4 , and also for all x-values. Ex) For  EMBED Equation.DSMT4 , find the following: a)  EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4  d)  EMBED Equation.DSMT4  Notice this resembles the problem above, except the first piece is slightly different. a)  EMBED Equation.DSMT4  b)  EMBED Equation.DSMT4  c)  EMBED Equation.DSMT4  since parts a) and b) are different values. We say the limit Does Not Exist (DNE) and there is a non-removable jump discontinuity. d)  EMBED Equation.DSMT4  Here the function value exists, while the limit does not. They dont necessarily have anything to do with each other, and can co-exist without the other. Can you draw an example where neither the limit nor the function values exist at a point? Need help? Here are a few examples that combine some skills: Examples) No Calculator Given the two functions f and h, such that  EMBED Equation.3  and  EMBED Equation.3  Find all the zeros of f. Find the value of p so that the function h is continuous at  EMBED Equation.3 . Justify. Using the value of p found in part b), determine whether h is even, odd, or neither. Justify your answer. For  EMBED Equation.3 , find the value of c to make f continuous at  EMBED Equation.3 . If  EMBED Equation.3 , what values of a and b make f continuous at  EMBED Equation.3 ? For  EMBED Equation.DSMT4 , determine the constant c, such that  EMBED Equation.DSMT4  exists. The graph of  EMBED Equation.3  is given by Find a number b such that  EMBED Equation.3  is continuous in  EMBED Equation.3  but not in  EMBED Equation.3 . Find numbers a and b such that  EMBED Equation.3 is continuous in (a,b) but not in (a,b]. Find the least number a such that  EMBED Equation.3 is continuous in  EMBED Equation.3 . Find  EMBED Equation.DSMT4  (Hint: break the function down using its piece-wise definition.) Find  EMBED Equation.DSMT4  Functions that are continuous over specified intervals are very important to us as students and lovers of mathematics because of their utility in answering the questions that arise from real-world problems. An important property of these continuous functions is called the Intermediate Value Theorem, or IVT for short. Now, any time we come across a theorem with its own name (and Im not talking about a name like Theorem 1.3 or Fred,) you can expect it to be very important. By merely invoking the name of the Theorem, you should know both the hypothesis, the conclusion, and how to apply the theorem. The IVT is no exception. Here it is: The Intermediate Value Theorem for continuous functions: If a function  EMBED Equation.DSMT4  is continuous on a closed interval  EMBED Equation.DSMT4 , then it takes on every value between  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Put differently, if we know of a y-value, say  EMBED Equation.DSMT4 , that resides between the y-values of the two endpoints,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , then we are guaranteed at least one x-value,  EMBED Equation.DSMT4 , between the endpoints that generates that y-value such that  EMBED Equation.DSMT4 .  For the theorem above to apply, the if part, known as the hypothesis, must be met before we can apply the consequence or conclusion of the theorem designated by the then part. Ex1) Prove that the function  EMBED Equation.DSMT4  must have a root on the interval  EMBED Equation.DSMT4 . Step one: Is the function continuous? Yes, since it is a polynomial, and because polynomials are continuous for all real numbers. Step two: Is the function continuous on the closed interval? Yes, since it continuous everywhere. Note that even if the interval was given to us on an open interval, we must still check for continuity on the closed interval!! Step three: Since the theorem applies, we now must consider the function values at the endpoints:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Step four: Now we make a concluding statement as follows: Since a root of  EMBED Equation.DSMT4 exists when  EMBED Equation.DSMT4 , and since  EMBED Equation.DSMT4 , By the IVT,  EMBED Equation.DSMT4  must have at least one root in the interval  EMBED Equation.DSMT4 . Ex2) Heres an example from the 2007 AP exam. FR #3:   When working with limits, you should also become adroit and adept at using limits of generic functions to find new limits of new functions created from combinations and modifications to those generic functions. Ill show you what I mean, but first, some important properties of limits that make it all work. 1.  EMBED Equation.DSMT4  2.  EMBED Equation.DSMT4  3. Heres the cool one:  EMBED Equation.DSMT4  4. Similarly:  EMBED Equation.DSMT4  5.  EMBED Equation.DSMT4  6.  EMBED Equation.DSMT4  , etc. . . In other words, we can move the limit anywhere in an expression to where the function part is, evaluate it, then combine the results in the prescribed manner. This is a rare feat in mathematics. Enjoy it. Lets try it out: Given the graphs of f and g are given below.   graph of f graph of g Determine whether the following limits exist. If they do, then find the limit. 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 j)  EMBED Equation.3 k)  EMBED Equation.3 l)  EMBED Equation.3  m)  EMBED Equation.3 n)  EMBED Equation.3 o)  EMBED Equation.3 p) 5- EMBED Equation.3  You should also master finding limits of specific functions without the aid of a calculator, a graph, or your brilliant friend sitting next to you. Heres a list of some strictly Algebraic Methods for evaluating limits. The names are not important, but recognizing the situations in which each method arises IS important. Direct Substitution This is the method one should ALWAYS try first. In the case whereby direct substitution leads to the indeterminate form  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 , this method works (for the most part.) This method works when the function is continuous at that point. Remember to try it first! By the way, all subsequent methods all lead back to direct substitution, as you shall see. Factor and Cancel Weve already seen this method. This is an ideal method to use when the function is a rational function of containing two polynomial functions. A  EMBED Equation.DSMT4  in this case guarantees that the bad guy factor, the one causing the zero in the denominator, can be divided out with its common factor in the numerator, leaving a function that paves over the hole in the original graph at that point. This allows us to use direct substitution on the remaining function. Ex 1)  EMBED Equation.DSMT4  Ex 2)  EMBED Equation.DSMT4  HINT: Synthetic Division can help you find the remaining factors when you know a root Ex 3)  EMBED Equation.DSMT4 =? Knowing a VA exists Remember any time direct sub yields  EMBED Equation.DSMT4 , there is a VA at that x-value Ex 1)  EMBED Equation.DSMT4  RATCON This method works well in the  EMBED Equation.DSMT4  case when there is a sum or difference of one or more radical terms. It involves multiplying by a clever form of one that is the  EMBED Equation.DSMT4  of the radical binomial. Ex 1)  EMBED Equation.DSMT4  **NOTICE that we leave the denominator in factored form so that we can divide out the bad guy factor. Ex 2)  EMBED Equation.DSMT4 ? LCM/LCM This method works well in the  EMBED Equation.DSMT4  case when there is a complex or compound fraction (a fraction within a fraction.) It involves multiplying by a clever form on one that involves the Least Constant Multiple of all the denominators in the fraction. Ex 1)  EMBED Equation.DSMT4  **NOTICE that we again leave the denominator in factored form so that we can divide out the bad guy factor. Ex 2)  EMBED Equation.DSMT4 =? Ex 3)  EMBED Equation.DSMT4 ? Sledgehammer or Expand and Cancel and Factor and Cancel . . . . This method works well in the  EMBED Equation.DSMT4  case when there is a some obvious math to do, like expanding a binomial or distributing factors. Usually expanding everything out and collecting like terms will lead to something in which the bad guy factor can then be removed. Ex 1)  EMBED Equation.DSMT4  Ex 2)  EMBED Equation.DSMT4 =? Getting Triggy Wit It This method involves cleverly rewriting trig expressions as equivalent ones that are easier to work with. Its basically the reason we spent time in precal doing trig proofs. This method requires memorizing two important limits that will repeatedly appear in many problems. Youll have fun doing a little numeric and algebraic gymnastics with these. MEMORIZE the following:  EMBED Equation.DSMT4  AND  EMBED Equation.DSMT4  Ex. 1)  EMBED Equation.DSMT4  Ex 2)  EMBED Equation.DSMT4 ? Ex 3)  EMBED Equation.DSMT4  Ex 4)  EMBED Equation.DSMT4 = Ex 5)  EMBED Equation.DSMT4  Ex 6)  EMBED Equation.DSMT4 = Ex 7)  EMBED Equation.DSMT4 = General Cleverness: These are when all else fails. What can you do to an expression without changing its values? Multiply by a clever form of one, add a clever form of zero, make equivalent substitutions (like trig identities,  EMBED Equation.DSMT4  with its piecewise definition or  EMBED Equation.DSMT4 ), and anything else you can think of. Ex1)  EMBED Equation.DSMT4 = Ex2)  EMBED Equation.DSMT4 = Ex3)  EMBED Equation.DSMT4 = A Limit Sandwich: Also called the Squeeze Theorem or Sandwich Theorem If  EMBED Equation.DSMT4  for all  EMBED Equation.DSMT4  in some interval containing c, AND if  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  This is a pretty common sense, and tasty, theorem. Basically if one function is above another one (or equal to it) all the time except maybe at one point, and if the to functions approach the same y-value at that point, then any function that is contained within them must also be sandwiched or squeezed through the same y-value at that point. Heres a visual   Another way to thing about it is that h and g are the bread, and f is the salami. I know what youre saying, or thinking: How and the heck do I go about finding such functions that bound the one Im interested in? Where do we get the bread? Thats a great question! Sometimes, they can be difficult to find. Other time, they magically appear through algebraic manipulations. In fact, weve already done this. Heres and example:  EMBED Equation.DSMT4  Here, finding the desired limit of  EMBED Equation.DSMT4  at zero, we stumbled across two other functions that contain our desired function that both have limits of one; therefore, we know our desired limit must be one. Heres the graph.  Here, we didnt really need to know what was going on graphically behind the scenes. The Squeeze Theorem worked its magic surreptitiously. Generally, though, youll be asked to find the limit of a function at a point, and you will be given information about the ones bounding it. Heres an example: Ex) If  EMBED Equation.DSMT4 , find  EMBED Equation.DSMT4 . Into the Great Beyond: LIMITS AT INFINITY Up to now, weve been finding limits at a single point, a specific x-value. Theres one last important task when evaluating limits, and that is checking them on the very, bitter ends of a graph, that is, for a function EMBED Equation.DSMT4 , we want to know  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The end behavior of a graph is very important, especially when trying to graph it by hand or answering test questions related to end behavior. Assuming a graph DOES have end behavior (unlike  EMBED Equation.DSMT4  at  EMBED Equation.DSMT4 ), there are only a couple of things that can happen. The graph can increase without bound (like  EMBED Equation.DSMT4 )  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  The graph can decrease without bound (like  EMBED Equation.DSMT4 )  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  The graph can oscillate between two fixed values (like  EMBED Equation.DSMT4 )  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  The graph will taper off towards a specific y-value  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  The last case is of particular interest to most math students. Heres why: Definition:  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4  if and only if there exists a Horizontal Asymptote (HA) at  EMBED Equation.DSMT4  Horizontal Asymptotes are a special quality of many Rational Functions (a polynomial over a polynomial.) There are 3 cases based on the comparison of the degrees of the leading terms of the numerator and denominator. Remember the leading terms are not necessarily the ones written in front, but rather the terms with the largest power of x. When taking limits at infinity, were essentially looking for the existence or non-existence of any HAs. We can do this by analyzing the effective growth rates of the numerator and denominator. Here are the three cases: Case 1: The degree of denominator is greater by any amount than the numerators. In this case, for extremely large values of x, even larger than that, the denominator is getting bigger faster than the numerator, so the fractions are getting smaller, and smaller, and smaller, and smaller, approaching nothing, or ZERO. The other terms in the fractions are insignificantly small in terms of contributing to the growth, and are only there to assure us that the values will only APPROACH and not EQUAL zero. In this case, there will ALWAYS be an HA at  EMBED Equation.DSMT4 . Ex)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  (but from the negative side) Case 2: The degree of the numerator and denominator are exactly the same. In this case, the top and bottom are essentially growing at the same rate. Again the trailing terms are insignificantly small compared to the leading terms. This means that the fractions or ratios are approaching the ratio of the leading terms, or the leading terms coefficients!! In this case, there will always be an HA at  EMBED Equation.DSMT4 . Ex)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Case 3: The degree of the numerator is greater by any amount than the denominator. In this case, the numerators growth is outpacing the denominators. The fractions or ratios are growing, and growing, and growing, forever, either in the positive or negative directions. The limit will not exist and there will be no HA. However, there will still be an asymptote that will take the equation of the quotient (degree difference), such as a Slant (Linear), Quadratic, Cubic asymptote, but unless you are working on your math Ph.D. or just a math geek, you will likely never have to worry about them, We will look at Slant Asymptotes, though. Ex)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  (to determine what type of infinity it will be, plug in a representative huge value into the leading terms and look what the resulting sign will be.) Ex)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Slant Asymptotes: When the degree of the numerator is one larger than the denominator, there is an SA at  EMBED Equation.DSMT4  as long as the remainder is NOT ZERO (what have we then?) Ex)  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , but there is an SA. We find it by long division:  EMBED Equation.DSMT4  Limits at infinity: Bonus Problems! Different functions grow at different rates for large values: log functions grow slowly, polynomials grow faster by order of degree, exponential growth functions grow faster than a polynomial of any degree, factorials are next, and the king of growth is  EMBED Equation.DSMT4  If you have a ratio of these types of functions, you can find limits at infinity by analyzing respective growth rates. Ex1)  EMBED Equation.DSMT4  Ex2)  EMBED Equation.DSMT4  Ex3)  EMBED Equation.DSMT4  Ex4)  EMBED Equation.DSMT4  Ex5)  EMBED Equation.DSMT4  Ex6)  EMBED Equation.DSMT4  Ex7)  EMBED Equation.DSMT4  Ex8)  EMBED Equation.DSMT4  Ex9)  EMBED Equation.DSMT4  Ex10)  EMBED Equation.DSMT4  Ex11)  EMBED Equation.DSMT4  Ex12)  EMBED Equation.DSMT4  Which of the above functions have Horizontal Asymptotes??? Before we close out limits altogether, lets summarize a bit: The limit is a y-value The limit can exist independently of a function value The limit only exists if the two one-sided limits BOTH exist and are the SAME value. When the left-sided limit equals the right-sided limit and both equal the function value, the function is said to be continuous at that point. If we have our calculator and dont know how to evaluate the limit, we can always graph it and plug values in on either side of our target value. When evaluating limits algebraically, indeterminate forms such as  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4  require us to roll up our sleeves and get to work. The limit may or may not exist. Piecewise functions are the exception to every rule. Our friend  EMBED Equation.DSMT4  and his family members will always have a jump discontinuity at the x-value that yields  EMBED Equation.DSMT4 . Evaluating if from either side of this involves plugging in ANY value from that side and evaluating, since the graph is composed two horizontal lines. A Limit can fail to exist at a point in any of the following circumstances: A jump discontinuity  EMBED Equation.DSMT4  A Vertical Asymptote  EMBED Equation.DSMT4  Graph not defined from both sides Ex)  EMBED Equation.DSMT4  Oscillation between two fixed values Ex)  EMBED Equation.DSMT4  Oscillation between two increasing large values Ex)  EMBED Equation.DSMT4  We can squeeze functions through a point even if the function doesnt exist there. Ex)  EMBED Equation.DSMT4  Limits form the basis of both differential and integral Calculus Limits are fun     Korpi Pg  PAGE 11 of  NUMPAGES 16  EMBED Equation.DSMT4  A function  EMBED Equation.DSMT4 is continuous at a point  EMBED Equation.DSMT4 if 1.  EMBED Equation.DSMT4  is defined 2.  EMBED Equation.DSMT4  exists 3.  EMBED Equation.DSMT4  Definition: If  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 , then there exists a vertical asymptote at  EMBED Equation.DSMT4 . Definition: A function  EMBED Equation.DSMT4 is continuous on an open interval  EMBED Equation.DSMT4  if it is continuous at each and every point in that interval. A function EMBED Equation.DSMT4 is continuous on a closed interval  EMBED Equation.DSMT4  if it is continuous on the open interval  EMBED Equation.DSMT4  and if the endpoints exhibit the following: i)  EMBED Equation.DSMT4  and ii)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  In this graph,  EMBED Equation.DSMT4  for all x except possibly at  EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , we know that  EMBED Equation.DSMT4  too!  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  So there is a HA at  EMBED Equation.DSMT4 . 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