For Use Only in 2013 2014– Pilot Program 96 Chapter 2 r Limits 46–47. Steep secant lines b. Create a graph that gives a more complete representation of f. a. Given the graph of f in the following figures, find the slope of the 1 2 1 1 22 y secant line that passes through 0, 0 and h, f h in terms of h, for h 7 0 and h 6 0. 20 b. Evaluate the limit of the slope of the secant line found in part (a) as h S 0+ and h S 0-. What does this tell you about the line tangent to the curve at 10, 02? 15 1 2 = 1>3 46. f x x ϭ 2000 10 y y 50 ϩ 100 x2 5 (h, h1/3) (0, 0) Ϫ4 Ϫ2 420 x h x y ϭ x1/3 Technology Exercises T 49–56. Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. x2 - 3x + 2 2>3 1 2 = 1 2 = - 2 1 2 = 47. f x x 49. f x 50. g x 2 ln x x10 - x9 y x p 1 2 = e 1 2 = a xb ͉ ͉ 6 y ϭ x2/3 51. h x 52. p x sec , for x 2 1x + 123 2 (h, h2/3) pu p 53. g1u2 = tan a b 54. q1s2 = (0, 0) 10 s - sin s h x 1 2 = 1 1 2 = 1>x 55. f x 56. g x e 1x sec x T 57. Can a graph intersect a vertical asymptote? A common mis- conception is that the graph of a function never intersects its verti- cal asymptotes. Let T 48. Care with graphing The figure shows the graph of the function 1 2 = 2000 3 - 4 * 3 4 4 6 f x graphed in the window 4, 4 0, 20 . if x 1 2 1 2 = - 50 + 100x f x W x 1 2 Ú . 1 2 1 2 1 2 x if x 1 a. Evaluate lim f x , lim f x , and lim f x . xS0+ xS0- xS0 Explain why x = 1 is a vertical asymptote of the graph of f and show that the graph of f intersects the line x = 1. QUICK CHECK ANSWERS 1. Answers will vary, but all graphs should have a vertical asymptote at x = 2. 2. - ∞; ∞ 3. As x S -4+ , x 6 0 and 1 + 2 7 1 + 2 S x 4 0, so x x 4 0 through negative values. 1x - 121x - 22 = 1 - 2 = 4. lim lim x 1 1, which is not xS2 x - 2 xS2 an infinite limit, so x = 2 is not a vertical asymptote. ➤ 2.5 Limits at Infinity Limits at infinity—as opposed to infinite limits—occur when the independent variable becomes large in magnitude. For this reason, limits at infinity determine what is called the end behavior of a function. An application of these limits is to determine whether a sys- tem (such as an ecosystem or a large oscillating structure) reaches a steady state as time increases. Copyright © 2014 Pearson Education, Inc. For Use Only in 2013 2014– Pilot Program 2.5 Limits at Infinity 97 y lim f (x) q Limits at Infinity and Horizontal Asymptotes xᠬϱ 1 2 = -1 1- ∞ ∞2 q Consider the function f x tan x, whose domain is , (Figure 2.30). As S ∞ 1 2 p> x becomes arbitrarily large (denoted x ), f x approaches 2, and as x becomes Horizontal Ϫ1 S - ∞ 1 2 -p> f (x) tan x arbitrarily large in magnitude and negative (denoted x ), f x approaches 2. asymptote These limits are expressed as - p - p 2 x lim tan 1 x = and lim tan 1 x = - . xS∞ 2 xS-∞ 2 Horizontal asymptote The graph of f approaches the horizontal line y = p>2 as x S ∞, and it approaches the Ϫq horizontal line y = -p>2 as x S - ∞. These lines are called horizontal asymptotes. lim f (x) q xᠬϪϱ FIGURE 2.30 DEFINITION Limits at Infinity and Horizontal Asymptotes 1 2 y If f x becomes arbitrarily close to a finite number L for all sufficiently large and posi- lim f (x) L tive x, then we write xᠬϱ L 1 2 = lim f x L. S f (x) y ϭ f (x) x ∞ Ϫϱ 1 2 = x ៬ We say the limit of f x as x approaches infinity is L. In this case, the line y L is a horizontal asymptote of f (Figure 2.31). The limit at negative infinity, x x ϱ 1 2 = lim f x M, is defined analogously. When the limit exists, the horizontal ៬ xS - ∞ f (x) asymptote is y = M. M lim f (x) M xᠬϪϱ x ➤ FIGURE 2.31 QUICK CHECK 1 Evaluate x>1x + 12 for x = 10, 100, and 1000. What is lim ? xS∞ x + 1 EXAMPLE 1 Limits at infinity Evaluate the following limits. a + 10 b a + 3 sin x b a. lim 2 2 b. lim 3 ➤ The limit laws of Theorem 2.3 and the xS - ∞ x xS∞ 1x Squeeze Theorem apply if x S a is SOLUTION replaced with x S ∞ or x S - ∞. a. As x becomes large and negative, x2 becomes large and positive; in turn, 10>x2 y approaches 0. By the limit laws of Theorem 2.3, 10 10 10 f (x) 2 a + b = + a b = + = x2 lim 2 2 lim 2 lim 2 2 0 2. xS-∞ x xS-∞ (+xS+)++*-∞ x c equals 2 equals 0 lim f (x) 2 lim f (x) 2 10 xᠬϪϱ xᠬϱ Notice that lim a2 + b is also equal to 2. Therefore, the graph of y = 2 + 10>x2 xS∞ x2 = S ∞ S - ∞ 2 approaches the horizontal asymptote y 2 as x and as x (Figure 2.32). y ϭ 2 b. The numerator of sin x> 1x is bounded between -1 and 1; therefore, for x 7 0, x 2 1 sin x 1 - … … . FIGURE 2.32 1x 1x 1x As x S ∞, 1x becomes arbitrarily large, which means that -1 1 lim = lim = 0. xS∞ 1x xS∞ 1x sin x It follows by the Squeeze Theorem (Theorem 2.5) that lim = 0. xS∞ 1x Copyright © 2014 Pearson Education, Inc. For Use Only in 2013 2014– Pilot Program 98 Chapter 2 r Limits y Using the limit laws of Theorem 2.3, 6 3 sin x sin x lim a3 + b = lim 3 + 3 lim a b = 3. 3 sin x S∞ 1 S∞ S∞ 1 5 f (x) ϭ 3 ϩ x x x (+x +)++*x x e equals 3 equals 0 4 3 sin x 3 The graph of y = 3 + approaches the horizontal asymptote y = 3 as x 1x 2 lim f (x) 3 becomes large (Figure 2.33). Note that the curve intersects its asymptote infinitely x ϱ ➤ 1 ៬ many times. Related Exercises 9–14 0 0 1020 30 40 x Infinite Limits at Infinity FIGURE 2.33 It is possible for a limit to be both an infinite limit and a limit at infinity. This type of limit 1 2 occurs if f x becomes arbitrarily large in magnitude as x becomes arbitrarily large in lim f (x) magnitude. Such a limit is called an infinite limit at infinity and is illustrated by the func- xᠬϱ y 1 2 = 3 tion f x x (Figure 2.34). DEFINITION Infinite Limits at Infinity 1 1 2 f (x) ϭ x3 If f x becomes arbitrarily large as x becomes arbitrarily large, then we write 1 2 = ∞ lim f x . xS∞ Ϫ1 1 x 1 2 = - ∞ 1 2 = ∞ 1 2 = - ∞ The limits lim f x , lim f x , and lim f x are S∞ S - ∞ S - ∞ Ϫ x x x 1 defined similarly. Infinite limits at infinity tell us about the behavior of polynomials for large-magnitude lim f (x) 1 2 = n values of x. First, consider power functions f x x , where n is a positive integer. xᠬϪϱ Figure 2.35 shows that when n is even, lim xn = ∞, and when n is odd, lim xn = ∞ S { ∞ S∞ FIGURE 2.34 and lim xn = - ∞. x x xS - ∞ ϭ 5 n 0 even: y n 0 odd: y y x n n n lim x lim x lim x y ϭ x7 x ៬ Ϯϱ x ៬ ϱ x ៬ Ϫϱ 60 y ϭ x6 y ϭ x4 20 y ϭ x3 40 Ϫ3 Ϫ2 1 2 3 x 20 Ϫ20 y ϭ x2 Ϫ3 Ϫ2 Ϫ1 1 2 3 x FIGURE 2.35 Copyright © 2014 Pearson Education, Inc. For Use Only in 2013 2014– Pilot Program 2.5 Limits at Infinity 99 1 2 = > n = -n It follows that reciprocals of power functions f x 1 x x , where n is a positive integer, behave as follows: 1 = -n = 1 = -n = lim n lim x 0 and lim n lim x 0. xS∞ x xS∞ xS - ∞ x xS - ∞ S { ∞ QUICK CHECK 2 Describe the behavior From here, it is a short step to finding the behavior of any polynomial as x . Let 1 2 = n + n - 1 + g + 2 + + of p1x2 = -3x3 as x S ∞ and as p x anx an - 1x a2x a1x a0. We now write p in the equivalent x S - ∞. ➤ form a - a - a 1 2 = n ° + n 1 + n 2 + g+ 0 ¢ p x x an 2 n . x x "x e e S0 S0 S0 Notice that as x becomes large in magnitude, all the terms in p except the first term ap- S {∞ 1 2 ≈ n proach zero.