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Unformatted text preview: 62. If and are positive numbers, prove that the equation has at least one solution in the interval . 63. Show that the function is continuous on . 64. (a) Show that the absolute value function is contin- uous everywhere. (b) Prove that if is a continuous function on an interval, then so is . (c) Is the converse of the statement in part (b) also true? In other words, if is continuous, does it follow that is continuous? If so, prove it. If not, find a counterexample. 65. A Tibetan monk leaves the monastery at 7:00 AM and takes his usual path to the top of the mountain, arriving at 7:00 P M . The following morning, he starts at 7:00 AM at the top and takes the same path back, arriving at the monastery at 7:00 P M . Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days. f f f f F x x , f x x 4 sin 1 x if x if x 1, 1 a x 3 2 x 2 1 b x 3 x 2 b a 55. Prove that is continuous at if and only if 56. To prove that sine is continuous, we need to show that for every real number . By Exercise 55 an equivalent statement is that Use (6) to show that this is true. 57. Prove that cosine is a continuous function. 58. (a) Prove Theorem 4, part 3. (b) Prove Theorem 4, part 5. 59. For what values of is continuous? 60. For what values of is continuous? Is there a number that is exactly 1 more than its cube? 61. t x x if x is rational if x is irrational t x f x 1 if x is rational if x is irrational f x lim h l sin a h sin a a lim x l a sin x sin a lim h l f a h f a a f LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES In Sections 2.2 and 2.4 we investigated infinite limits and vertical asymptotes. There we let approach a number and the result was that the values of became arbitrarily large (positive or negative). In this section we let become arbitrarily large (positive or nega- tive) and see what happens to . Let’s begin by investigating the behavior of the function defined by as becomes large. The table at the left gives values of this function correct to six decimal places, and the graph of has been drawn by a computer in Figure 1. As grows larger and larger you can see that the values of get closer and closer to 1. In fact, it seems that we can make the values of as close as we like to 1 by taking sufficiently large. This situation is expressed symbolically by writing lim x l x 2 1 x 2 1 1 x f x f x x x 1 y y=1 y= ≈-1 ≈+1 FIGURE 1 f x f x x 2 1 x 2 1 f y x y x 2.6 130 | | | | CHAPTER 2 LIMITS AND DERIVATIVES x 1 0.600000 0.800000 0.882353 0.923077 0.980198 0.999200 0.999800 0.999998 1000 100 50 10 5 4 3 2 1 f x SECTION 2.6 LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES | | | | 131 In general, we use the notation to indicate that the values of become closer and closer to as becomes larger and larger....
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- Spring '09