3.5 Limits at Infinity 195

3.5 Limits at Infinity

Determine (finite) limits at infinity. Determine the horizontal asymptotes, if any, of the graph of a function. Determine infinite limits at infinity.

Limits at Infinity y This section discusses the “end behavior” of a function on an infinite interval. Consider 3x2 f(x) = the graph of 4 x2 + 1 3x 2 f ͑x͒ ϭ x 2 ϩ 1 2 f(x) → 3 f(x) → 3 ͑ ͒ as x → −∞ as x → ∞ as shown in Figure 3.32. Graphically, you can see that the values of f x appear to approach 3 as x increases without bound or decreases without bound. You can come to x −4 −3 −2 −11234 the same conclusions numerically, as shown in the table. The limit of ͑x͒ as xf approaches Ϫϱ or ϱ is 3. x decreases without bound. x increases without bound.

Figure 3.32

x Ϫϱ → Ϫ100 Ϫ10 Ϫ1 0 1 10 100 → ϱ f͑x͒ 3 → 2.9997 2.9703 1.5 0 1.5 2.9703 2.9997 → 3

f ͑x͒ approaches 3. f ͑x͒ approaches 3.

The table suggests that the value of f ͑x͒ approaches 3 as x increases without bound ͑x → ϱ͒. Similarly,f ͑x͒ approaches 3 as x decreases without bound ͑x → Ϫϱ͒. These limits at infinity are denoted by

lim f ͑x͒ ϭ 3 Limit at negative infinity x→Ϫϱ REMARK The statement and lim f ͑x͒ ϭ L or x→Ϫϱ lim f ͑x͒ ϭ 3. Limit at positive infinity lim f ͑x͒ ϭ L means that the x→ϱ x→ϱ limit exists and the limit is To say that a statement is true as x increases without bound means that for some equal to L. (large) real number M, the statement is true for all x in the interval ͭx: x > Mͮ. The next definition uses this concept.

Definition of Limits at Infinity Let L be a real number. y 1. The statement lim f ͑x͒ ϭ L means that for each ␧ > 0 there exists an x→ϱ lim f(x) = L M > 0 such that Խf ͑x͒ Ϫ LԽ < ␧ whenever x > M. x→∞ 2. The statement lim f ͑x͒ ϭ L means that for each ␧ > 0 there exists an x→Ϫϱ N < 0 such that Խf͑x͒ Ϫ LԽ < ␧ whenever x < N. ε L ε The definition of a limit at infinity is shown in Figure 3.33. In this figure, note that x M for a given positive number ␧, there exists a positive number M such that, for x > M, the graph of f will lie between the horizontal lines f͑x͒ is within ␧ units of L as x → ϱ. Figure 3.33 y ϭ L ϩ␧ and y ϭ L Ϫ␧.

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 196 Chapter 3 Applications of Differentiation Horizontal Asymptotes Exploration In Figure 3.33, the graph of f approaches the line y ϭ L as x increases without bound. Use a graphing utility to The line y ϭ L is called a horizontal asymptote of the graph of f. graph 2x 2 ϩ 4x Ϫ 6 f ͑x͒ ϭ . 3x 2 ϩ 2x Ϫ 16 Definition of a Horizontal Asymptote The line y ϭ L is a horizontal asymptote of the graph of f when Describe all the important ͑ ͒ ϭ ͑ ͒ ϭ features of the graph. Can lim f x L or lim f x L. x→Ϫϱ x→ϱ you find a single viewing window that shows all of these features clearly? Note that from this definition, it follows that the graph of a function of x can have Explain your reasoning. at most two horizontal asymptotes—one to the right and one to the left. What are the horizontal Limits at infinity have many of the same properties of limits discussed in asymptotes of the graph? Section 1.3. For example, if lim f͑x͒ and lim g͑x͒ both exist, then How far to the right do you x→ϱ x→ϱ have to move on the graph lim ͓ f ͑x͒ ϩ g͑x͔͒ ϭ lim f ͑x͒ ϩ lim g͑x͒ →ϱ →ϱ →ϱ so that the graph is within x x x 0.001 unit of its horizontal and asymptote? Explain your lim ͓ f ͑x͒g͑x͔͒ ϭ ͓ lim f ͑x͔͓͒ lim g͑x͔͒. reasoning. x→ϱ x→ϱ x→ϱ Similar properties hold for limits at Ϫϱ. When evaluating limits at infinity, the next theorem is helpful.

THEOREM 3.10 Limits at Infinity If r is a positive rational number and c is any real number, then

c ϭ lim 0. x→ϱ x r Furthermore, if x r is defined when x < 0, then

c ϭ lim 0. x→Ϫϱ x r A proof of this theorem is given in Appendix A. See LarsonCalculus.com for Bruce Edwards’s video of this proof.

Finding a Limit at Infinity

2 y Find the limit: lim ΂5 Ϫ ΃. x→ϱ x 2 10 Solution Using Theorem 3.10, you can write 8 f(x) = 5 − 2 2 2 x2 lim ΂5 Ϫ ΃ ϭ lim 5 Ϫ lim Property of limits 6 x→ϱ x 2 x→ϱ x→ϱ x 2 ϭ Ϫ 4 5 0 ϭ 5. So, the line y ϭ 5 is a horizontal asymptote to the right. By finding the limit x −6 −4 −2 2 4 6 2 lim ΂5 Ϫ ΃ Limit as x → Ϫϱ. x→Ϫϱ x2 y ϭ 5 is a horizontal asymptote. you can see that y ϭ 5 is also a horizontal asymptote to the left. The graph of the Figure 3.34 function is shown in Figure 3.34.

Finding a Limit at Infinity

2x Ϫ 1 Find the limit: lim . x→ϱ x ϩ 1 Solution Note that both the numerator and the denominator approach infinity as x approaches infinity.

lim ͑2x Ϫ 1͒ → ϱ x→ϱ 2x Ϫ 1 lim x→ϱ x ϩ 1 lim ͑x ϩ 1͒ → ϱ x→ϱ

ϱ This results in ϱ, an indeterminate form. To resolve this problem, you can divide both REMARK When you the numerator and the denominator by x. After dividing, the limit may be evaluated encounter an indeterminate form as shown. such as the one in Example 2, 2x Ϫ 1 you should divide the numerator 2x Ϫ 1 x and denominator by the highest lim ϭ lim Divide numerator and denominator by x. x→ϱ x ϩ 1 x→ϱ x ϩ 1 power of x in the denominator. x 1 2 Ϫ y ϭ x lim Simplify. x→ϱ 1 6 1 ϩ x 5 Ϫ 1 4 − lim 2 lim f(x) = 2x 1 x→ϱ x→ϱ x x + 1 ϭ Take limits of numerator and denominator. 3 1 lim 1 ϩ lim x→ϱ x→ϱ x 1 2 Ϫ 0 ϭ Apply Theorem 3.10. x 1 ϩ 0 −5 −4 −3 −2 123 −1 ϭ 2 So, the line y ϭ 2 is a horizontal asymptote to the right. By taking the limit as x → Ϫϱ, y ϭ 2 is a horizontal asymptote. you can see that y ϭ 2 is also a horizontal asymptote to the left. The graph of the Figure 3.35 function is shown in Figure 3.35.

TECHNOLOGY You can test the reasonableness of the limit found in Example 2 by evaluating f ͑x͒ for a few large positive values of x. For instance, f ͑100͒ Ϸ 1.9703, f ͑1000͒ Ϸ 1.9970, 3 and f ͑10,000͒ Ϸ 1.9997. Another way to test the reasonableness of the limit is to use a graphing utility. For instance, in Figure 3.36, the graph of 2x Ϫ 1 f ͑x͒ ϭ 0 80 x ϩ 1 0 is shown with the horizontal line y ϭ 2. Note As x increases, the graph of f moves ϭ that as x increases, the graph of f moves closer closer and closer to the line y 2. and closer to its horizontal asymptote. Figure 3.36

A Comparison of Three Rational Functions

See LarsonCalculus.com for an interactive version of this type of example. Find each limit. 2x ϩ 5 2x 2 ϩ 5 2x 3 ϩ 5 a.lim b.lim c. lim x→ϱ 3x 2 ϩ 1 x→ϱ 3x 2 ϩ 1 x→ϱ 3x 2 ϩ 1 Solution In each case, attempting to evaluate the limit produces the indeterminate form ϱ͞ϱ. a. Divide both the numerator and the denominator by x 2. 2x ϩ 5 ͑2͞x͒ ϩ ͑5͞x 2͒ 0 ϩ 0 0 lim ϭ lim ϭ ϭ ϭ 0 x→ϱ 3x 2 ϩ 1 x→ϱ 3 ϩ ͑1͞x 2͒ 3 ϩ 0 3 b. Divide both the numerator and the denominator by x 2. 2x 2 ϩ 5 2 ϩ ͑5͞x 2͒ 2 ϩ 0 2 lim ϭ lim ϭ ϭ → 2 ϩ →ϱ ϩ ͑ ͞ 2͒ ϩ MARIA GAETANA AGNESI x ϱ 3x 1 x 3 1 x 3 0 3 (1718–1799) c. Divide both the numerator and the denominator by x 2. Agnesi was one of a handful of women to receive credit for 2x 3 ϩ 5 2x ϩ ͑5͞x 2͒ ϱ lim ϭ lim ϭ significant contributions to x→ϱ 3x 2 ϩ 1 x→ϱ 3 ϩ ͑1͞x 2͒ 3 mathematics before the twentieth century. In her early twenties, she You can conclude that the limit does not exist because the numerator increases wrote the first text that included without bound while the denominator approaches 3. both differential and integral calculus. By age 30, she was an honorary member of the faculty Example 3 suggests the guidelines below for finding limits at infinity of rational at the University of Bologna. See LarsonCalculus.com to read functions. Use these guidelines to check the results in Example 3. more of this biography. For more information on the contributions of women to GUIDELINES FOR FINDING LIMITS AT ±ؕ OF RATIONAL mathematics, see the article “Why FUNCTIONS Women Succeed in Mathematics” by Mona Fabricant, Sylvia Svitak, 1. If the degree of the numerator is less than the degree of the denominator, then and Patricia Clark Kenschaft in the limit of the rational function is 0. Mathematics Teacher.To view this 2. If the degree of the numerator is equal to the degree of the denominator, then article, go to MathArticles.com. the limit of the rational function is the ratio of the leading coefficients. 3. If the degree of the numerator is greater than the degree of the denominator, then the limit of the rational function does not exist.

y The guidelines for finding limits at infinity of rational functions seem reasonable when you consider that for large values of x, the highest-power term of the rational 2 f(x) = 1 function is the most “influential” in determining the limit. For instance, x2 + 1 1 lim x→ϱ x2 ϩ 1 x x is 0 because the denominator overpowers the numerator as increases or decreases −2 −112without bound, as shown in Figure 3.37. lim f(x) = 0 lim f(x) = 0 The function shown in Figure 3.37 is a special case of a type of curve studied by x → −∞ x → ∞ the Italian mathematician Maria Gaetana Agnesi. The general form of this function is f has a horizontal asymptote at y ϭ 0. 3 Figure 3.37 8a f ͑x͒ ϭ Witch of Agnesi x 2 ϩ 4a 2 and, through a mistranslation of the Italian word vertéré, the curve has come to be known as the Witch of Agnesi. Agnesi’s work with this curve first appeared in a comprehensive text on calculus that was published in 1748. The Granger Collection, New York

In Figure 3.37, you can see that the function 1 f͑x͒ ϭ x2 ϩ 1 approaches the same horizontal asymptote to the right and to the left. This is always true of rational functions. Functions that are not rational, however, may approach different horizontal asymptotes to the right and to the left. This is demonstrated in Example 4.

A Function with Two Horizontal Asymptotes

Find each limit. 3x Ϫ 2 3x Ϫ 2 a.lim b. lim x→ϱ Ί2x 2 ϩ 1 x→Ϫϱ Ί2x 2 ϩ 1 Solution a. For x > 0, you can write x ϭ Ίx 2. So, dividing both the numerator and the denominator by x produces Ϫ 3x 2 Ϫ 2 Ϫ 2 Ϫ 3 3 3x 2 ϭ x ϭ x ϭ x Ί2x 2 ϩ 1 Ί2x 2 ϩ 1 2x 2 ϩ 1 1 2 Ί Ί2 ϩ Ίx x 2 x 2 and you can take the limit as follows. 2 y 3 3 Ϫ y = , 3x Ϫ 2 x 3 Ϫ 0 3 2 lim ϭ lim ϭ ϭ Horizontal x→ϱ Ί 2 ϩ x→ϱ Ί ϩ Ί 4 2x 1 1 2 0 2 asymptote ϩ Ί2 2 to the right x b. For x < 0, you can write x ϭϪΊx 2. So, dividing both the numerator and the x denominator by x produces −6 −4 −224 3x Ϫ 2 2 2 3 Ϫ 3 Ϫ 3x − 2 3x Ϫ 2 x x x − 3 f(x) = ϭ ϭ ϭ y = , − 2 Ί 2 ϩ Ί 2 ϩ 2 4 2x + 1 2x 1 2x 1 2x 2 ϩ 1 1 2 ϪΊ ϪΊ2 ϩ Horizontal ϪΊx x 2 x 2 asymptote to the left and you can take the limit as follows. Functions that are not rational may 2 have different right and left horizontal 3 Ϫ 3x Ϫ 2 x 3 Ϫ 0 3 asymptotes. lim ϭ lim ϭ ϭϪ Figure 3.38 x→Ϫϱ Ί2x 2 ϩ 1 x→Ϫϱ 1 ϪΊ2 ϩ 0 Ί2 ϪΊ2 ϩ x 2 ͑ ͒ ϭ ͑ Ϫ ͒͞Ί 2 ϩ 2 The graph of f x 3x 2 2x 1 is shown in Figure 3.38.

TECHNOLOGY PITFALL If you use a graphing utility to estimate a limit, be sure that you also confirm the estimate analytically—the pictures shown by a

−8 8 graphing utility can be misleading. For instance, Figure 3.39 shows one view of the graph of 2x 3 ϩ 1000x 2 ϩ x −1 y ϭ . x 3 ϩ 1000x 2 ϩ x ϩ 1000 The horizontal asymptote appears to be the line y ϭ 1, but it is actually the line From this view, one could be convinced that the graph has y ϭ 1 as a horizontal y ϭ 2. asymptote. An analytical approach shows that the horizontal asymptote is actually Figure 3.39 y ϭ 2. Confirm this by enlarging the viewing window on the graphing utility.

In Section 1.3 (Example 9), you saw how the Squeeze Theorem can be used to evaluate limits involving trigonometric functions. This theorem is also valid for limits at infinity.

Limits Involving Trigonometric Functions

Find each limit. sin x a.lim sin x b. lim x→ϱ x→ϱ y x

y = 1 Solution x 1 a. As x approaches infinity, the sine function oscillates between 1 and Ϫ1. So, this f(x) = sin x limit does not exist. x b. Because Ϫ1 Յ sin x Յ 1, it follows that for x > 0, x π Ϫ1 Յ sin x Յ 1 lim sin x = 0 x x x x→∞ x where −1 y = − 1 x 1 1 lim ΂Ϫ ΃ ϭ 0 and lim ϭ 0. x→ϱ x x→ϱ x As x increases without bound, f͑x͒ approaches 0. So, by the Squeeze Theorem, you can obtain Figure 3.40 sin x lim ϭ 0 x→ϱ x as shown in Figure 3.40.

Oxygen Level in a Pond

Let f͑t͒ measure the level of oxygen in a pond, where f͑t͒ ϭ 1 is the normal (unpolluted) level and the time t is measured in weeks. When t ϭ 0, organic waste is dumped into the pond, and as the waste material oxidizes, the level of oxygen in the pond is t2 Ϫ t ϩ 1 f͑t͒ ϭ . t2 ϩ 1 What percent of the normal level of oxygen exists in the pond after 1 week? After 2 weeks? After 10 weeks? What is the limit as t approaches infinity? f(t) Solution When t ϭ 1, 2, and 10, the levels of oxygen are as shown. 1.00 2 Ϫ ϩ ͑ ͒ ϭ 1 1 1 ϭ 1 ϭ (2, 0.6) (10, 0.9) f 1 2 50% 1 week 0.75 1 ϩ 1 2 22 Ϫ 2 ϩ 1 3 t2 − t + 1 f͑2͒ ϭ ϭ ϭ 60% 2 weeks 0.50 f(t) = 2 ϩ (1, 0.5) t2 + 1 2 1 5 Oxygen level 10 2 Ϫ 10 ϩ 1 91 0.25 f͑10͒ ϭ ϭ Ϸ 90.1% 10 weeks 10 2 ϩ 1 101 t 246810 To find the limit as t approaches infinity, you can use the guidelines on page 198, or you Weeks can divide the numerator and the denominator by t 2 to obtain The level of oxygen in a pond t 2 Ϫ t ϩ 1 1 Ϫ ͑1͞t͒ ϩ ͑1͞t 2͒ 1 Ϫ 0 ϩ 0 approaches the normal level of 1 as lim ϭ lim ϭ ϭ 1 ϭ 100%. t→ϱ t 2 ϩ 1 t→ϱ 1 ϩ ͑1͞t 2͒ 1 ϩ 0 t approaches ϱ. Figure 3.41 See Figure 3.41.

Definition of Infinite Limits at Infinity REMARK Determining Let f be a function defined on the interval ͑a, ϱ͒. whether a function has an 1. The statement lim f ͑x͒ ϭ ϱ means that for each positive number M, there infinite limit at infinity is useful x→ϱ in analyzing the “end behavior” is a corresponding number N > 0 such that f ͑x͒ > M whenever x > N. of its graph. You will see 2. The statement lim f ͑x͒ ϭϪϱ means that for each negative number M, x→ϱ examples of this in Section 3.6 there is a corresponding number N > 0 such that f ͑x͒ < M whenever x > N. on curve sketching. Similar definitions can be given for the statements ͑ ͒ ϭ ͑ ͒ ϭϪ lim f x ϱ and lim f x ϱ. x→Ϫϱ x→Ϫϱ

Finding Infinite Limits at Infinity

y Find each limit.

3 a.lim x3 b. lim x3 x→ϱ x→Ϫϱ 2 f(x) = x3 Solution 1 a. As x increases without bound,x3 also increases without bound. So, you can write x lim x3 ϭ ϱ. −3 −23−1 1 2 x→ϱ −1 b. As x decreases without bound,x3 also decreases without bound. So, you can write −2 lim x3 ϭϪϱ. x→Ϫϱ − 3 The graph of f ͑x͒ ϭ x3 in Figure 3.42 illustrates these two results. These results agree with the Leading Coefficient Test for polynomial functions as described in Section P.3. Figure 3.42

Finding Infinite Limits at Infinity

Find each limit. y 2x2 Ϫ 4x 2x2 Ϫ 4x a.lim b. lim x→ϱ x ϩ 1 x→Ϫϱ x ϩ 1 2x2 − 4x 6 f(x) = x + 1 3 Solution One way to evaluate each of these limits is to use long division to rewrite x the improper rational function as the sum of a polynomial and a rational function. −12 −9 −6912−3 3 6 2 −3 2x Ϫ 4x 6 a. lim ϭ lim ΂2x Ϫ 6 ϩ ΃ ϭ ϱ y = 2x − 6 →ϱ ϩ →ϱ ϩ −6 x x 1 x x 1 2x2 Ϫ 4x 6 b. lim ϭ lim ΂2x Ϫ 6 ϩ ΃ ϭϪϱ x→Ϫϱ x ϩ 1 x→Ϫϱ x ϩ 1 The statements above can be interpreted as saying that as x approaches ±ϱ, the function f ͑x͒ ϭ ͑2x2 Ϫ 4x͒͑͞x ϩ 1͒ behaves like the function g͑x͒ ϭ 2x Ϫ 6. In Section 3.6, you will see that this is graphically described by saying that the line Figure 3.43 y ϭ 2x Ϫ 6 is a slant asymptote of the graph of f, as shown in Figure 3.43.

3.5 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

Matching In Exercises 1–6, match the function with one of Finding Limits at Infinity In Exercises 13 and 14, find the graphs [(a), (b), (c), (d), (e), or (f)] using horizontal lim hͧxͨ, if possible. x→ϱ asymptotes as an aid. 13.f ͑x͒ ϭ 5x3 Ϫ 3x2 ϩ 10x 14. f ͑x͒ ϭϪ4x2 ϩ 2x Ϫ 5 y y (a) (b) f͑x͒ f͑x͒ (a)h͑x͒ ϭ (a) h͑x͒ ϭ 3 3 x2 x 2 f͑x͒ f͑x͒ 1 (b)h͑x͒ ϭ (b) h͑x͒ ϭ x3 x2 1 x −3 −1 123 ͑ ͒ ͑ ͒ ͑ ͒ ϭ f x ͑ ͒ ϭ f x x (c)h x 4 (c) h x 3 −2 −112 x x − 1 −3 Finding Limits at Infinity In Exercises 15–18, find each (c)y (d) y limit, if possible. 3 3 x2 ϩ 2 3 Ϫ 2x 15. (a) lim 16. (a) lim 2 x→ϱ x 3 Ϫ 1 x→ϱ 3x 3 Ϫ 1 1 1 x2 ϩ 2 3 Ϫ 2x x x (b)lim (b) lim → 2 Ϫ → Ϫ −3 −2 −1123 123 x ϱ x 1 x ϱ 3x 1 −1 x2 ϩ 2 3 Ϫ 2x2 −2 (c)lim (c) lim → Ϫ → Ϫ −3 −3 x ϱ x 1 x ϱ 3x 1 5 Ϫ 2x3͞2 5x3͞2 y y 17. (a) lim 18. (a) lim (e) (f) x→ϱ 3x2 Ϫ 4 x→ϱ 4x2 ϩ 1 8 4 5 Ϫ 2x3͞2 5x3͞2 3 (b)lim 3͞2 (b) lim 3͞2 6 x→ϱ 3x Ϫ 4 x→ϱ 4x ϩ 1 2 4 5 Ϫ 2x3͞2 5x3͞2 1 (c)lim (c) lim x→ϱ Ϫ x→ϱ Ί ϩ 2 x 3x 4 4 x 1 x −31−2 −1 23 −6 −4 −224 Finding a Limit In Exercises 19–38, find the limit. −2 ϩ 3 5 Ϫ x 2 19.lim ΂4 ΃ 20. lim ΂ ΃ 2x 2x x→ϱ x x→Ϫϱ x 3 1.f͑x͒ ϭ 2. f͑x͒ ϭ 2 ϩ 2 x 2 Ίx ϩ 2 2x Ϫ 1 4x2 ϩ 5 2 21.lim 22. lim x x x→ϱ 3x ϩ 2 x→Ϫϱ x2 ϩ 3 3.f͑x͒ ϭ 4. f͑x͒ ϭ 2 ϩ 2 ϩ 4 ϩ x 2 x 1 x 5x3 ϩ 1 2 23.lim 24. lim 4 sin x 2x Ϫ 3x ϩ 5 x→ϱ x 2 Ϫ 1 x→ϱ 10x3 Ϫ 3x2 ϩ 7 5.f͑x͒ ϭ 6. f͑x͒ ϭ 2 ϩ 2 ϩ x 1 x 1 5x 2 x3 Ϫ 4 25.lim 26. lim x→Ϫϱ x ϩ 3 x→Ϫϱ x2 ϩ 1 Numerical and Graphical Analysis In Exercises 7–12, use a graphing utility to complete the table and estimate the x x 27.lim 28. lim limit as x approaches infinity. Then use a graphing utility to x→Ϫϱ Ίx 2 Ϫ x x→Ϫϱ Ίx 2 ϩ 1 graph the function and estimate the limit graphically. 2x ϩ 1 5x2 ϩ 2 29.lim 30. lim x→Ϫϱ Ίx 2 Ϫ x x→ϱ Ίx2 ϩ 3 0 1 2 3 4 5 6 x 10 10 10 10 10 10 10 Ίx2 Ϫ 1 Ίx4 Ϫ 1 31.lim 32. lim →ϱ Ϫ →Ϫϱ 3 Ϫ f ͑x͒ x 2x 1 x x 1 x ϩ 1 2x 33.lim ͞ 34. lim ͞ 4x ϩ 3 2x 2 x→ϱ ͑x2 ϩ 1͒1 3 x→Ϫϱ ͑x6 Ϫ 1͒1 3 7.f͑x͒ ϭ 8. f͑x͒ ϭ 2x Ϫ 1 x ϩ 1 1 1 35.lim 36. lim cos Ϫ6x 10 x→ϱ 2x ϩ sin x x→ϱ x 9.f͑x͒ ϭ 10. f ͑x͒ ϭ Ί4x 2 ϩ 5 Ί2x2 Ϫ 1 sin 2x x Ϫ cos x 37.lim 38. lim 1 3 x→ϱ x x→ϱ x 11.f͑x͒ ϭ 5 Ϫ 12. f ͑x͒ ϭ 4 ϩ x 2 ϩ 1 x2 ϩ 2

Horizontal Asymptotes In Exercises 39–42, use a graphing utility to graph the function and identify any WRITING ABOUT CONCEPTS (continued) horizontal asymptotes. 57. Using Symmetry to Find Limits If f is a continuous function such that lim f ͑x͒ ϭ 5, find, if possible, ԽxԽ Խ3x ϩ 2Խ x→ϱ 39.f ͑x͒ ϭ 40. f ͑x͒ ϭ lim f ͑x͒ for each specified condition. x ϩ 1 x Ϫ 2 x→Ϫϱ 3x Ί9x2 Ϫ 2 (a) The graph of f is symmetric with respect to the y -axis. 41.f ͑x͒ ϭ 42. f ͑x͒ ϭ Ίx2 ϩ 2 2x ϩ 1 (b) The graph of f is symmetric with respect to the origin.

Finding a Limit In Exercises 43 and 44, find the limit. t and find the limit as t → 0؉.ͨ 58. A Function and Its Derivative The graph of a function/1 ؍ Hint: Let xͧ f is shown below. To print an enlarged copy of the graph, go to 1 1 MathGraphs.com. 43.lim x sin 44. lim x tan x→ϱ x x→ϱ x y

Finding a Limit In Exercises 45–48, find the limit. (Hint: 6 Treat the expression as a fraction whose denominator is 1, and rationalize the numerator.) Use a graphing utility to verify 4 your result. 2 ͑ ϩ Ί 2 ϩ ͒ ͑ Ϫ Ί 2 ϩ ͒ f 45.lim x x 3 46. lim x x x x→Ϫϱ x→ϱ x 47.lim ͑3x ϩ Ί9x 2 Ϫ x ͒ 48. lim ͑4x Ϫ Ί16x 2 Ϫ x ͒ −4 −224 →Ϫϱ →ϱ x x −2 Numerical, Graphical, and Analytic Analysis In (a) Sketch fЈ. Exercises 49–52, use a graphing utility to complete the table ͑ ͒ Ј͑ ͒ and estimate the limit as x approaches infinity. Then use a (b) Use the graphs to estimate lim f x and lim f x . x→ϱ x→ϱ graphing utility to graph the function and estimate the limit. (c) Explain the answers you gave in part (b). Finally, find the limit analytically and compare your results with the estimates. Sketching a Graph In Exercises 59–74, sketch the graph of the equation using extrema, intercepts, symmetry, and x 100 101 102 103 104 105 106 asymptotes. Then use a graphing utility to verify your result. ͑ ͒ x x Ϫ 4 f x 59.y ϭ 60. y ϭ 1 Ϫ x x Ϫ 3 ͑ ͒ ϭ Ϫ Ί ͑ Ϫ ͒ ͑ ͒ ϭ 2 Ϫ Ί ͑ Ϫ ͒ x ϩ 1 2x 49.f x x x x 1 50. f x x x x x 1 61.y ϭ 62. y ϭ 2 Ϫ Ϫ 2 ϩ x 4 9 x ͑ ͒ ϭ 1 ͑ ͒ ϭ x 1 51. f x x sin 52. f x x 2 2x 2 2x xΊx 63.y ϭ 64. y ϭ x 2 ϩ 16 x 2 Ϫ 4 WRITING ABOUT CONCEPTS 65.xy 2 ϭ 9 66. x 2y ϭ 9 3x 3x Writing In Exercises 53 and 54, describe in your own 67.y ϭ 68. y ϭ words what the statement means. x Ϫ 1 1 Ϫ x 2 ͑ ͒ ϭ ͑ ͒ ϭ 3 1 53.lim f x 4 54. lim f x 2 69.y ϭ 2 Ϫ 70. y ϭ 1 Ϫ x→ϱ x→Ϫϱ x 2 x 2 4 55. Sketching a Graph Sketch a graph of a differentiable 71.y ϭ 3 ϩ 72. y ϭ ϩ 1 function f that satisfies the following conditions and has x x2 x ϭ 2 as its only critical number. x 3 x 73.y ϭ 74. y ϭ Ίx 2 Ϫ Ίx 2 Ϫ fЈ͑x͒ < 0 for x < 2 4 4 fЈ͑x͒ > 0 for x > 2 Analyzing a Graph Using Technology In Exercises lim f͑x͒ ϭ 6 75–82, use a computer algebra system to analyze the graph of x→Ϫϱ the function. Label any extrema and/or asymptotes that exist. lim f͑x͒ ϭ 6 →ϱ x 5 1 75.f͑x͒ ϭ 9 Ϫ 76. f͑x͒ ϭ 56. Points of Inflection Is it possible to sketch a graph of x 2 x 2 Ϫ x Ϫ 2 a function that satisfies the conditions of Exercise 55 and x Ϫ 2 x ϩ 1 has no points of inflection? Explain. 77.f͑x͒ ϭ 78. f͑x͒ ϭ x 2 Ϫ 4x ϩ 3 x 2 ϩ x ϩ 1

3x 2x 79.f͑x͒ ϭ 80. g͑x͒ ϭ Ί 2 ϩ Ί 2 ϩ 4x 1 3x 1 88. HOW DO YOU SEE IT? The graph shows the x 2 sin 2x temperature T, in degrees Fahrenheit, of molten 81.g͑x͒ ϭ sin΂ ΃, x > 3 82. f͑x͒ ϭ x Ϫ 2 x glass t seconds after it is removed from a kiln.

Comparing Functions In Exercises 83 and 84, (a) use a T graphing utility to graph f and g in the same viewing window, (0, 1700) (b) verify algebraically that f and g represent the same function, and (c) zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.) 72 t x3 Ϫ 3x2 ϩ 2 83. f͑x͒ ϭ x͑x Ϫ 3͒ (a) Find limϩ T. What does this limit represent? t→0 ͑ ͒ ϭ ϩ 2 g x x ͑ Ϫ ͒ (b) Find lim T. What does this limit represent? x x 3 t→ϱ x3 Ϫ 2x2 ϩ 2 (c) Will the temperature of the glass ever actually reach 84. f͑x͒ ϭϪ 2x2 room temperature? Why? 1 1 g͑x͒ ϭϪ x ϩ 1 Ϫ 2 x2 89. Modeling Data The average typing speeds S (in words 85. Engine Efficiency per minute) of a typing student after t weeks of lessons are The efficiency of an internal combustion engine is shown in the table.

1 Efficiency ͑%͒ ϭ 100 1 Ϫ t 5 1015202530 ΄ ͑ ͞ ͒c΅ v1 v2 S 28 56 79 90 93 94 ͞ where v1 v2 is the ratio of the uncompressed 100t2 gas to the compressed A model for the data is S ϭ , t > 0. 65 ϩ t2 gas and c is a positive constant dependent on (a) Use a graphing utility to plot the data and graph the model. the engine design. Find (b) Does there appear to be a limiting typing speed? Explain. the limit of the efficiency 90. Modeling Data A heat probe is attached to the heat as the compression ratio exchanger of a heating system. The temperature T (in degrees approaches infinity. Celsius) is recorded t seconds after the furnace is started. The results for the first 2 minutes are recorded in the table. 86. Average Cost A business has a cost of C ϭ 0.5x ϩ 500 for producing x units. The average cost per unit is t 0 15304560 C T 25.2° 36.9° 45.5° 51.4° 56.0° C ϭ . x Find the limit of C as x approaches infinity. t 75 90 105 120 87. Physics Newton’s First Law of Motion and Einstein’s T 59.6° 62.0° 64.0° 65.2° Special Theory of Relativity differ concerning a particle’s behavior as its velocity approaches the speed of light c. In the (a) Use the regression capabilities of a graphing utility to find graph, functions N and E represent the velocity v, with respect ϭ 2 ϩ ϩ a model of the form T1 at bt c for the data. to time t, of a particle accelerated by a constant force as (b) Use a graphing utility to graph T . predicted by Newton and Einstein, respectively. Write limit 1 statements that describe these two theories. (c) A rational model for the data is v 1451 ϩ 86t T ϭ . N 2 58 ϩ t c E Use a graphing utility to graph T2. ͑ ͒ ͑ ͒ (d) Find T1 0 and T2 0 .

(e) Find lim T2. t→ϱ (f) Interpret the result in part (e) in the context of the problem. t Is it possible to do this type of analysis using T1? Explain. Straight 8 Photography/Shutterstock.com

91. Using the Definition of Limits at Infinity The graph of 94. Using the Definition of Limits at Infinity Consider

2 ͑ ͒ ϭ 2x 3x f x lim . x2 ϩ 2 x→Ϫϱ Ίx2 ϩ 3 is shown. (a) Use the definition of limits at infinity to find values of N that correspond to ␧ϭ0.5. y (b) Use the definition of limits at infinity to find values of N that correspond to ␧ϭ0.1.

ε f Proof In Exercises 95–98, use the definition of limits at infinity to prove the limit.

1 ϭ 2 ϭ 95.lim 0 96. lim 0 x x→ϱ x2 x→ϱ Ίx x2 x1 1 ϭ 1 ϭ Not drawn to scale 97.lim 0 98. lim 0 x→Ϫϱ x3 x→Ϫϱ x Ϫ 2 ϭ ͑ ͒ (a) Find L lim f x . x→ϱ 99. Distance A line with slope m passes through the point ␧ (b) Determine x1 and x2 in terms of . ͑0, 4͒. (c) Determine M, where M > 0, such that Խ f ͑x͒ Ϫ LԽ < ␧ for (a) Write the shortest distance d between the line and the x > M. point ͑3, 1͒ as a function of m. (d) Determine N, where N < 0, such that Խ f ͑x͒ Ϫ LԽ < ␧ for (b) Use a graphing utility to graph the equation in part (a). x < N. ͑ ͒ ͑ ͒ (c) Find lim d m and lim d m . Interpret the results 92. Using the Definition of Limits at Infinity The graph of m→ϱ m→Ϫϱ geometrically. 6x f͑x͒ ϭ 100. Distance A line with slope m passes through the point Ίx2 ϩ 2 ͑0, Ϫ2͒. is shown. (a) Write the shortest distance d between the line and the point ͑4, 2͒ as a function of m. y (b) Use a graphing utility to graph the equation in part (a). ͑ ͒ ͑ ͒ ε (c) Find lim d m and lim d m . Interpret the results f m→ϱ m→Ϫϱ geometrically. 101. Proof Prove that if x x x 2 1 ͑ ͒ ϭ n ϩ . . . ϩ ϩ p x an x a1x a0 ε and

Not drawn to scale ͑ ͒ ϭ m ϩ . . . ϩ ϩ q x bm x b1x b0 ϭ ͑ ͒ ϭ ͑ ͒ (a) Find L lim f x and K lim f x . where a 0 and b 0, then x→ϱ x→Ϫϱ n m ␧ (b) Determine x1 and x2 in terms of . 0, n < m ͑ ͒ Ϫ ␧ (c) Determine M, where M > 0, such that Խ f x LԽ < for p͑x͒ an lim ϭ , n ϭ m. x > M. → ͑ ͒ x ϱ q x Άbm (d) Determine N, where N < 0, such that Խ f ͑x͒ Ϫ KԽ < ␧ for ±ϱ, n > m x < N. 102. Proof Use the definition of infinite limits at infinity to 93. Using the Definition of Limits at Infinity Consider 3 ϭ prove that lim x ϱ. x→ϱ 3x lim . x→ϱ Ίx2 ϩ 3 True or False? In Exercises 103 and 104, determine whether the statement is true or false. If it is false, explain (a) Use the definition of limits at infinity to find values of M why or give an example that shows it is false. that correspond to ␧ϭ0.5. 103. If fЈ͑x͒ > 0 for all real numbers x, then f increases without (b) Use the definition of limits at infinity to find values of M bound. that correspond to ␧ϭ0.1. 104. If fЉ͑x͒ < 0 for all real numbers x, then f decreases without bound.

Section 3.5 (page 202) 1. f 2. c 3. d 4. a 5. b 6. e 49.

7. 0 1 2 3 x 10 10 10 10 x 100 101 102 103 104 105 106 f ͑x͒ 7 2.2632 2.0251 2.0025 f ͑x͒ 1.000 0.513 0.501 0.500 0.500 0.500 0.500

2 x 104 105 106 ͓ Ί ͑ ͔͒ 1 ͑ ͒ lim x x x 1 2 f x 2.0003 2.0000 2.0000 −1 8 x→

4x 3 −2 lim 2 → −10 10 x 2x 1 51. x 100 101 102 103 104 105 106 −10 f ͑x͒ 0.479 0.500 0.500 0.500 0.500 0.500 0.500

9. 1 x 100 101 102 103 The graph has a hole at x 0. f ͑x͒ 2 2.9814 2.9998 3.0000 −2 2 1 1 lim x sin x→ 2x 2 x 104 105 106 −1 f ͑x͒ 3.0000 3.0000 3.0000

10 53. As x becomes large,f͑x͒ approaches 4. 6x 55. Answers will vary. Sample answer: Let lim 3 − x→ Ί 2 6 10 10 4x 5 f͑x͒ 6. 0.1͑x 2͒2 1 y −10 11. 8 x 100 101 102 103 ͑ ͒ f x 4.5000 4.9901 4.9999 5.0000 4

x 104 105 106 x −2 642 f ͑x͒ 5.0000 5.0000 5.0000 57. (a) 5 (b) 5 6 59. y 61. y 1 lim 5 5 4 4 → ΂ 2 ΃ x x 1 3 3 2 2 1 1 − 1 8 x x 0 1 5324 −43−1 2 4 −1 13. (a) (b) 5 (c) 0 15. (a) 0 (b) 1 (c) −2 −2 17. (a) 0 (b)2 (c) 19. 4 21.2 23. 0 −3 −3 3 3 − 1 4 −4 25. 27.1 29.2 31.2 33. 35. 0 37. 0 63. y 65. y 4 39. 41. 6 2 4 y = 1 y = 3 3 y = −1 1 2 −6 6 −99 1 x x −8−6 −4 −2 42 68 4321 765 y = −3 −1 −1 −2 −4 −6 −3 1 −2 −4 43. 1 45. 0 47. 6

67. y 69. y 83. (a) 8 (c) 70 7 4 6 3 − 5 2 fg= 80 80 4 1 − 3 4 8 x 2 −4 −3 −2 2 34 −2 −70 x −4−3 −2 −1 1 23456 (b) Proof Slant asymptote: y x −2 85. 100% 87. lim N͑t͒ ; lim E͑t͒ c t→ t→ 71. y 73. y 100 89. (a) 120 (b) Yes. lim S 100 8 20 t→ 1 7 16 6 12 5 8 4 4 3 x 2 −5 −4 −3 −2 −1 21 345 − 5 30 8 0 x −12 − − − − 4 3 2 1 123 54 −16 − 91. (a) lim f͑x͒ 2 20 x→ 12 2 75. y = 9 77. 4 2 4 2 x = 0 x = 3 (b) x1 Ί , x2 Ί −1 5 4 2 4 2 y = 0 (c) Ί (d) Ί −66 x = 1 −2 −2 5Ί33 93. (a) Answers will vary. M 95–97. Proofs 11 79. 2 81. 2π (π − 2 , 1 ( Ί 1.2 29 177 y = 3 (b) Answers will vary. M 2 59 − 3 3 y = − 3 Խ3m 3Խ 2 99. (a) d͑m͒ y = sin(1) Ίm2 1 −2 (b) 6 (c) lim d͑m͒ 3; 3 12 m→ 0 lim d͑m͒ 3; m→ ± −12 12 As m approaches , the distance approaches 3 −2

101. Proof 2x 103. False. Let f͑x͒ . f͑x͒ > 0 for all real numbers. Ίx2 2