Horizontal asymptotes

Transcription

1 Roberto s Notes on Differential Calculus Chapter : 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 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 illuminating eamples. We say that a quantity q approaches negative infinity, and write q, when that quantity can be made less than any predetermined negative value. Warning bells The symbols and - represent concepts, NOT numbers. Therefore usual algebraic rules do NOT apply to them. Definition We say that a quantity q approaches infinity, and write q, when that quantity can be made larger than any predetermined positive value. 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! Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page

2 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 If f( ) the graph eits towards a corner of the window, as shown in the picture. Knot on your finger f( ) f( ) f( ) f( ) Since and - are not numbers, the following technical points must be kept in mind. 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. 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. In it notation: This is reflected in the graph of the function, which eists towards to top left and top right corners. Enough jargon. Let s see the graphical interpretations of such its. Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page

3 Eample: y 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 eists the window towards the top right corner. 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. Finite its at infinity tend to be more interesting. Eample: y sin As becomes large, y also becomes large and with the same sign, since sin is always a small number, bounded between - and. Therefore: sin ; sin This is shown by the graph of the function. 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. Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page

4 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 f is to 0. Therefore, 0 is a horizontal asymptote, both on the left and on the right and we have: 0. Eample: Here again we can let y become as close to 0 as we want by letting become sufficiently large, so that 0 is a vertical asymptote: 0 This time the function is approaching the asymptote from different sides, a fact reflected in the graph, but that does not affect the fact that the same line is a horizontal asymptote on both sides.. Is the ais the only possible horizontal asymptote Definitely not, but we need to consider more methods to compute its before we can clearly identify more horizontals asymptotes. However, the ais will show up very frequently in our eamples, because of the following fact. Strategy for its at infinity: The it law of gravity If a function is of the form for large values of, Law of gravity a p b f( ) p ( ) q ( ) and: p is bounded, that is for some finite values a and b; q then f( ) 0 and 0 is a right horizontal asymptote for the function. The same occurs as for left horizontal asymptotes. Another nickname! The idea is again that since the denominator becomes large without bound, but the top is bounded, the whole fraction can be made as small as we want, squashed down to 0 by its own weight, so to speak. Eample: sin As becomes large, the numerator stays between - and, but the denominator becomes, obviously, as large as we want. By the law of gravity, we can conclude that sin 0. Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page 4

5 Therefore, the line 0 is a right horizontal asymptote for this function. Of course the same happens to the left. Notice that this last eample also brings up, with even more force, something that I pointed out, perhaps less convincingly, in connection to vertical asymptotes. Warning bells Asymptotes are connected to its, NOT with lack of intersections: a function CAN cross its horizontal asymptotes, even infinitely many times! The following strategy is a direct consequence of the law of gravity. Strategy for computing its at infinity: Divide top and bottom by the highest power of If a function is of the form compute both p and f f( ) p ( ) q ( ), to it may be effective to divide q by the largest power of that appears in the function and then computing the resulting it. Proof Implementing this strategy can change the function to a ratio of epressions containing several fractions, to each of which we can apply the law of gravity. 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. Eample: If we divide top and bottom by the highest power, that is The variable now only appears n the denominator, so that by applying the law of gravity to each of the fractions, they all approach 0 and we can conclude that: , we obtain: Therefore 5 is a horizontal asymptote, as supported by the graph. This strategy can also be used when the highest power is not really a power, but an eponential. Eample: y tanh If we use the definition of the hyperbolic tangent, we have: Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page 5

6 e tanh e e e But for large values of, e is very large, while e becomes as small as we want, so that we can try to divide top and bottom by e, thus getting: e e tanh e e e e By applying the law of gravity we conclude that tanh. On the other hand, for large negative values of, the roles of the two eponentials is reversed and we can try to divide top and bottom by getting: e, thus tanh e e This is confirmed by the graph and shows that a function can have different asymptotes on the two sides. Because of the way this strategy works, that is, einating all the lower powers by moving them to the denominator, a quick and dirty way to implement it has another cute name. Strategy for its at infinity The law of the jungle If a function is of the form compute f to retain only the terms of f( ) p ( ) q ( ), to it is sufficient, in many cases, p and those of q that generate the largest value in the numerator and denominator respectively. In other words, the largest term wins and leads in the computation. Eample: If we einate all powers lower than, we are left with : Notice that this strategy is stated in fairly vague terms and it applies in many cases only. So, use it with care and, especially at the beginning, check whether it is appropriate to use it or whether there are complications that make it incorrect. Also, notice that it works only when the numerator and denominator are made up of a sum of terms, rather than other combinations of functions. Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page 6

7 Eample: 56 4 The highest power in this epression is, since the in the top is under a root. So, to find the right horizontal asymptote, we einate all lower power terms and obtain: With can do the same when looking for the left horizontal asymptote, but we must be careful, since generally that when is negative, :, so The graph confirms that this is another function with different horizontal asymptotes on the left and the right. 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 of the it laws, be they formal or cute, make sure to follow proper algebra! Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page 7

8 Learning questions for Section -5 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!. 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. What is the value of 5. What is the value of sin 6. What is the value of arctan 8. What is the value of e 9. What is the value of e 0. What is the value of ln. What is the value of ln. What is the left horizontal asymptote of f tanh. What is the right horizontal asymptote of f tanh 7. What is the value of arctan Computation questions: Determine both its at infinity for each of the following functions and interpret your conclusions in terms of the graph. In particular, identify any horizontal asymptotes Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page 8

9 5. cos 6. e 5 7. yln( ) tanh e tanh e cosh Theory questions:. How do we identify horizontal asymptotes. Which graphical feature occurs when f( ) eists and is finite 4. When a horizontal asymptote occurs in a rational function, what does the fraction s denominator usually approach 5. Do the functions sin and y cos have horizontal asymptotes. Can polynomial functions have horizontal asymptotes 6. Why does the function f arcsin not have a horizontal asymptote Proof questions:. Show that if a rational function has a horizontal asymptote, it is the same one on both sides. What questions do you have for your instructor Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page 9

10 Differential Calculus Chapter : Limits and continuity Section 5: Horizontal asymptotes Page 0