Roberto s Notes on Differential Calculus Chapter 1: Limits and continuity Section 5 Limits at infinity and Horizontal asymptotes What you need to know already: The concept, notation and terminology of its. What you can learn here: Limit at infinity and their interpretation, especially in relation to horizontal asymptotes. When computing a it we ask what happens to the values f of a function when the independent variable approaches a certain value c. But equally interesting is the question of what happens to the function when becomes large without bounds, be it in the positive or negative direction. To put the issue in a visual contet, when we use a calculator to sketch the graph of a function, we only see the portion of the graph that shows up in the chosen window. But what happens to f when become very large and goes outside any window we may choose? Can we tell from it what happens to this function outside of the window we see? It looks like the function has some kind of periodic behavior as far as we can see, but will it hold? A larger window produces this new graph: Eample: A certain function produces this graph in the window to -5 to 5: It still looks oscillating, but again, what if we make it larger? This is another situation where the graphical approach is too weak and we need to devise a better method. To properly address this issue, we need to clarify some concepts and terminology about which many students have dangerous misconceptions. So, bear with me as I provide them before giving you some more illuminating eamples. Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 1
Definition We say that a quantity q approaches infinity, and write q, when q can be made larger than any predetermined positive value. We say that a quantity q approaches negative infinity, and write q, when q can be made less than any predetermined negative value. Let s see the graphical interpretations of such its, starting from infinite its at infinity. Knot on your finger If f( ) the graph eits towards a corner of the window, as shown in this picture. Warning bells The symbols and - represent concepts, NOT numbers. Therefore usual algebraic rules do NOT apply to them. f( ) f( ) f( ) f( ) You said that already in the last section, in relation to the law of balloons! Correct, and I will probably state that again, as it is an important concept that many students forget from time to time! A it of the form at infinity. A it of the form Definition infinite it at infinity. f f c is called a it is called an Eample: y Here we notice that by letting become sufficiently large, either in positive or negative values, we can make y as large as we want. In it notation: This is reflected in the graph of the function, which eits towards the top left and top right corners. Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page
Eample: ln We know that the natural logarithm is only defined for positive values of and that it becomes increasingly large in positive values as becomes large. Therefore: ln DNE ln Notice that the second it des not eist either, as a it, but our notation eplains that the graph eits the window towards the top right corner. sin is always a small number, bounded between -1 and 1. Therefore: sin ; sin This is shown by the graph of the function. What we just observed is a general concept that should be clear in your mind and used properly. Knots on your finger Limits at infinity, be they finite or infinite, make sense only if the domain of the function etends to the appropriate type of infinity. Otherwise they do not eist. An infinite it at infinity does not eist as a it; its notation and terminology are meant to provide additional information about why the it does not eist, rather than state its eistence. But the way the graphs go to infinity, when they do, is still unpredictable! You are correct, and this is another point worth remembering. Knot on your finger An infinite it at infinity only tells us the general direction in which the graph eits the window, NOT how fast it does so or with what pattern. More calculus methods are needed to obtain such information. Eample: sin Let us now look at finite its at infinity, which tend to be more interesting. As becomes large, y also becomes large and with the same sign, since Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page
If f ( ) a Definition, where a is a finite number, the line y a is called a right horizontal asymptote for the function f. If f ( ) a, where a is a finite number, the line y a is called a left horizontal asymptote for the function f. Eample: Here we can let y become as close to 0 as we want by letting become sufficiently large, so that 0 is a horizontal asymptote: 0 But this time the graph is below the asymptote on the left and above on the right. That does not affect the fact that the same line is a horizontal asymptote on both sides. Eample: Here we notice that by letting become as large we want, we can make y become as close to 0 as we want. The larger becomes, the closer Therefore: 0 f is to 0. and 0 is a horizontal asymptote, both on the left and on the right.. In the last eample the graph of the function is above the asymptote on both sides, but this need not be true always. Is the ais the only possible horizontal asymptote? Definitely not: all you have to do is shift the function vertically to have it somewhere else. And there are also functions that have different horizontal asymptotes on the two sides! Eample: e ytanh e e e e For large positive values of, becomes negligible (we ll see shortly how to make this observation more formal) and we can ignore it in the formula, thus concluding that: e e e tanh 1 e e e For large negative values of, it is e that becomes negligible, so that: e e e tanh 1 e e e This is clearly confirmed by a calculator s graph, as shown here: Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 4
Eample: sin As becomes large, the numerator stays between -1 and 1, but the denominator becomes, obviously, as large as we want. By the law of sin gravity, we can conclude that 0. Therefore, the line 0 is a right horizontal asymptote for this function. Of course the same happens to the left. Having clarified that any horizontal line can be the horizontal asymptote for some function, it is true that the ais will show up more frequently in our eamples. This is not only because 0 is a nice, simple number that can make our computations easier, but also because of the it law of gravity that I mentioned in an earlier section. If a function is of the form Technical fact p ( ) f( ), where, as q ( ), p is bounded and q, then f( ) 0 and 0 is a right horizontal asymptote for the function. The same occurs as for left horizontal asymptotes. Wait a minute! This cannot be an asymptote: the graph touches it repeatedly! I see that it is time for an important reminder. Warning bells Asymptotes are related to its, NOT to the absence of intersections with the graph of the function. A function CAN cross its horizontal asymptotes, even infinitely many times! The following strategy is a direct consequence of the law of gravity and can be used effectively in many situations that you will encounter frequently. To continue helping you in remembering these special it strategies and facts, I have given a cute name to this one too. Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 5
Strategy for its at infinity: The law of the jungle p ( ) To compute it may be effective to divide q ( ) both p and q by a term that is largest, as, among those used in the function, and then computing the resulting it. of gravity to each of the fractions, they all approach 0 and we conclude that: 81 1 0 0 1 4 5 0 0 5 5 Therefore 1 is a horizontal asymptote, as shown by the graph. 5 Proof Implementing this strategy can change the function to a ratio of epressions containing several fractions, to most of which we can apply the law of gravity, while the remaining ones are constant. Although this method does not always work, it is effective in many situations and it is therefore worth keeping in mind and trying when appropriate. Notice that this function also intersects its horizontal asymptote (just before ), this time only once. Eample: 81 4 5 This may look like a difficult it to compute, because of the compleity of its formula, but it becomes very simple if we divide top and bottom by the highest power, that is. This leads us to a different form of the function that may look even more complicated, but that is easier to analyze in terms of its it at infinity: 8 1 1 81 4 5 4 5 The variable now only appears in denominators, so that by applying the law This strategy can also be used when the largest term is not a power, but an eponential. For instance, we can use it to confirm the horizontal asymptotes of the hyperbolic tangent function we saw earlier. Eample: e ytanh e Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 6 e e For large values of, e is very large, while e 0, so that we can divide top and bottom by e, thus getting: 1 1 e e tanh e e e 1 1 e
We can now apply the law of gravity we conclude that 1 0 tanh 1. 1 0 On the other hand, for large negative values of, the roles of the eponentials e is reversed and we can divide top and bottom by, thus getting: e 1 0 1 tanh 1 e 1 0 1 Do you call it the law of the jungle because the strongest term wins? Eactly! Notice how this strategy works: it allows us to einate the smaller terms by moving them all to denominators. This leads to a quick and dirty way to implement it that you may have seen in high school, but that needs to be used carefully. Strategy for implementing the law of the jungle When the law of the jungle is considered for use, it may be possible to do so by simply retaining the largest terms of the function and ignoring the rest. This method works always for rational functions, but must be used with great care for other types of functions. Of course, this is the same conclusion we reached earlier, but we did it more quickly. Eample: 56 4 The largest term in this function is, since the in the top is under a root. So, to find the right horizontal asymptote, we einate all smaller terms and obtain: 4 5 6 We can do the same when looking for the left horizontal asymptote, but we must be careful, since generally : 5 6 4 The graph confirms that this is another function with different horizontal asymptotes on the left and the right., so that when is negative, Eample: 81 4 5 If we einate all powers lower than, we are left with: 8 1 1 1 4 5 5 5 5 I like the idea of einating smaller terms, but can we always figure out which one is the largest term? Ecellent point and the answer, unfortunately, is no! That is why I stated earlier that this method works always, and I might add easily, for rational functions, but not in general. In fact a warning is appropriate. Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 7
Warning bells The epression largest term is very vague and, while it can be interpreted easily when dealing with just powers (polynomials) or just eponentials, it can become very dangerous when dealing with more complicated mitures of terms and factors. There are other situations that can occur where any of the strategies we have seen here are combined with other methods to arrive at the proper it value. But we need to develop some more computational techniques before we can look at them. So, for now, if you see something in a it at infinity that sounds suspicious, wait until you have learned those further methods. In an earlier section we saw the Squeeze Theorem, which tells us that a function squeezed between two other functions that have the same it at a value c will also have the same it there. Well, that fact holds true also for its at infinity. However, do not forget the ited applicability of this theorem. For completeness, for your reference and as a reminder, here is the theorem in the contet of its at infinity. Technical fact: then g Similarly, if: then L. whenever c, f g f 1 and f L f L 1 g L. The two situations are visualized in the following two graphs. L g f f1 g The Squeeze Theorem Given functions,, f f g and numbers c 1 and L with the properties that: whenever c, f g f 1 and f L f L 1 g f f1 g L Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 8
Summary Limits at infinity tell us how the graph eits the field of view, also called the window. If a it at infinity is finite, it indicates the presence of a horizontal asymptote. Horizontal asymptotes are related to its at infinity, NOT to intersections or lack thereof. A horizontal asymptote may touch the graph, even infinitely many times. Common errors to avoid When using it methods, be they formal or cute, make sure to follow proper algebra! Remember that the Law of the jungle must be used with care, as it is easy to ignore terms that are, in fact, important. Learning questions for Section D 1-5 Review questions: 1. Describe what is meant by a it at infinity.. Eplain how to interpret the presence of infinite its at infinity.. Eplain how to interpret the presence of finite its at infinity. 4. Describe how to identify and interpret the presence of horizontal asymptotes for a function. 5. Both vertical and horizontal asymptotes are defined in terms of its and of infinity. Describe what generates the difference between them in such terms. Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 9
Memory questions: Please notice that this set of memory questions includes facts related to its at infinity that you learned in relation to the corresponding functions, not in this section. Consider them as a strong hint that memorizing these facts is important! 1. How many horizontal asymptotes can a function have?. Which two its are computed when searching for a horizontal asymptote?. Which eponential functions have a right horizontal asymptote? 4. 1 What is the value of? 5. sin What is the value of? 6. What is the value of arctan? 8. What is the value of e 9. What is the value of e 10. What is the value of ln? 11. What is the value of ln? 1. What is the left horizontal asymptote of f tanh? 1. What is the right horizontal asymptote of f tanh??? 7. What is the value of arctan? Computation questions: For each of the functions presented in questions 1-4, determine both its at infinity and interpret your conclusions in relation to the graph of the function. In particular, identify any horizontal asymptotes. 1.. 5 4. 4. 4 1 4 1 5. 6. 1 1 Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 10
7. 8. 9. 10. 11. 1. 1. 14. 5 10 6 1 10 4 cos cos e 1 y e y 5 e 1 ln 15. f sin ln 16. yln(1 ) 5 18. 9 5 19. 1 0. 1 4 4 8 1. 6 5. 5. 4. 5. 4 6 6 10 4 4 4 8 4 1 7. 8. 9. 4 5 7 tanh e tanh e cosh 0. sinh cosh 1... 4. cos sin ln sinh cos sin ln cosh tan 7 sin tan 7 cos 17. 1 5 4 5 6. 1 9 11 Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 11
5. If f is a function such that its it as approaches infinity? e f 4 tanh, what is 6. If a function f is such that for any 5, cos f 1 e, what is its it at infinity? Theory questions: 1. How do we identify horizontal asymptotes?. What is the smallest number of horizontal asymptotes that a function can have?. What is the largest number of horizontal asymptotes that a function can have? 4. Which graphical feature occurs when f( ) eists and is finite? 5. Can polynomial functions have horizontal asymptotes? 6. When a horizontal asymptote occurs in a rational function, what does the fraction s denominator usually approach? 7. Do the functions sin and cos have horizontal asymptotes? 8. Why does the function f sin 9. Why does not eist? ln 10. Is it true that if in the range of 11. Why is it impossible for f? arcsin not have a horizontal asymptote? is a horizontal asymptote for f? f, then is not to be a horizontal asymptote for a function Proof questions: sin 1. Use valid arguments to determine the value of, or to eplain why it ln does not eist. As usual, arguments based only on calculator work are not sufficient.. Show that if a rational function has a horizontal asymptote, it is the same one on both sides. sin. Determine the value of ln reasons for your conclusions. and clearly eplain the mathematical Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 1
4. Compare the following three functions: 1 f ; g 1 ; c 1 a) Do they have the same horizontal asymptotes? What does that say about their behavior for large values of? b) Do they have the same vertical asymptotes? What does that say about their behavior for small values of? 5. It is known that 1 e. Use this fact, other information about 1 sin y y basic its, and some appropriate algebra, to show that y0. Templated questions: 1. Compute both its at infinity of any function on which you are working and interpret their graphical meaning. What questions do you have for your instructor? Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 1
Differential Calculus Chapter 1: Limits and continuity Section 5: Horizontal asymptotes Page 14