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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Sullivan 35. f is continuous at c = 0. 36. f is discontinuous at c = 0. 37. f is continuous at c = π 4 . 38. f is discontinuous at c = 1. 39. f is continuous on { x| x ≠ 4}. 40. f is continuous on { x| x ≠ 0} 41. f is continuous on ⎧⎪ 3π ⎫⎪ ⎨xx≠ + 2 kπ ⎬ , k any integer. ⎩⎪ 2 ⎭⎪ 42. f is continuous on the set of all real numbers. 43. f is continuous on { x| x > 0, x ≠3}. 44. f is continuous on the set of all real numbers. 45. f is continuous on the set of all real numbers. 46. f is continuous on the set of all real numbers. 47. 0 48. 0 49. 0 50. 0 51. See TSM. 52. See TSM. 53. See TSM. 54. See TSM. 55. See TSM. 56. See TSM. 57. See TSM. 58. f (0) =π 59. f (0) = π, f (1) = π 60. Yes. 61. Yes. 62. See TSM. 63. See TSM. 64. See TSM. 65. Explanations will vary. For sample explanation, see TSM. 66. Explanations will vary. For sample explanation, see TSM. 67. See TSM. 68. 0 69. See TSM. 70. See TSM. 126 Chapter 1 • Limits and Continuity In Problems 35–38, determine whether f is continuous at the number c. ⎧ ⎨ 3 cos x if x < 0 35. f (x) = 3 if x = 0 at c = 0 ⎩ x + 3 if x > 0 ⎧ ⎨ cos x if x < 0 36. f (x) = 0 if x = 0 at c = 0 ⎩ e x if x > 0 ⎧ ⎨ sin θ if θ ≤ π 4 37. f (θ) = ⎩ cos θ if θ> π at c = π 4 4 tan −1 x if x < 1 38. f (x) = at c = 1 ln x if x ≥ 1 In Problems 39–46, determine where f is continuous. x 2 − 4x x 2 − 5x + 1 39. f (x) = sin x − 4 40. f (x) = cos 2x 41. 1 1 f (θ) = 42. f (θ) = 1 + sin θ 1 + cos 2 θ 43. f (x) = ln x x − 3 44. f (x) = ln(x 2 + 1) PAGE 125 45. f (x) = e −x sin x 46. e x f (x) = 1 + sin 2 x Applications and Extensions In Problems 47–50, use the Squeeze Theorem to find each limit. 47. lim x 2 sin 1 48. lim x 1 − cos 1 x→0 x x→0 x 49. lim x 2 1 − cos 1 1 50. lim x x→0 x + 3x 2 sin x→0 x In Problems 51–54, show that each statement is true. sin(ax) 51. lim x→0 sin(bx) = a cos(ax) ; b = 0 52. lim b x→0 cos(bx) = 1 sin(ax) 53. lim = a x→0 bx b ; b = 0 1 − cos(ax) 54. lim = 0; a = 0, b = 0 x→0 bx 1 − cos x 55. Show that lim x→0 x 2 = 1 2 . 56. Squeeze Theorem If 0 ≤ f (x) ≤ 1 for every number x, show that lim[x 2 f (x)] = 0. x→0 57. Squeeze Theorem If 0 ≤ f (x) ≤ M for every x, show that lim x→0 [x2 f (x)] = 0. sin(π x) 58. The function f (x) = is not defined at 0. Decide how to x define f (0) so that f is continuous at 0. sin(π x) 59. Define f (0) and f (1) so that the function f (x) = x(1 − x) is continuous on the interval [0, 1]. ⎧ ⎨ sin x if x = 0 60. Is f (x) = x continuous at 0? ⎩ 1 if x = 0 ⎧ ⎨ 1 − cos x if x = 0 61. Is f (x) = x continuous at 0? ⎩ 0 if x = 0 1 62. Squeeze Theorem Show that lim x n sin = 0, where n x→0 x is a positive integer. (Hint: Look first at Problem 56.) 63. Prove lim sin θ = 0. θ→0 (Hint: Use a unit circle as shown in the figure, first assuming 0
Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4 Section 1.5 • Infinite Limits; Limits at Infinity; Asymptotes 127 AP® Practice Problems ⎧ ⎨ sin(2x) if x = 0 PAGE 121 1. The function g(x) = 2x ⎩ k if x = 0 is continuous at x = 0. What is the value of k? 1 (A) 0 (B) (C) 1 (D) 2 2 PAGE sin(4x) 121 2. lim x→0 2x (A) 0 = (B) 1 2 PAGE 125 3. The function f (x) = (C) 1 (D) 2 x 3 + 2x 2 if x ≤−2 e 2x+4 if x > −2 . Find lim f (x) if it exists. x→−2 (A) 0 (B) 1 (C) 16 (D) The limit does not exist. PAGE 1 − cos 2 (3x) 121 4. lim x→0 x 2 = (A) 0 (B) 1 (C) 3 (D) 9 PAGE 125 5. Which of the following functions are continuous for all real numbers x? I. f (x) = x 1/3 II. g(x) = sec x III. h(x) = e −x (A) I only (B) I and II only (C) I and III only (D) I, II, and III RECALL The symbols ∞ (infinity) and −∞ (negative infinity) are not numbers. The symbol ∞ expresses unboundedness in the positive direction, and −∞ expresses unboundedness in the negative direction. PAGE 1 121 6. Find lim if it exists. x→0 x csc x (A) −1 (B) 0 (C) 1 (D) The limit does not exist. sin x − π PAGE 121 7. lim 3 x→π/3 x − π = 3 (A) − π π (B) 0 (C) 1 (D) 3 3 PAGE 1 − cos x 122 8. lim x→0 3 sin 2 = x (A) 1 6 PAGE 125 9. If f (x) = (B) 1 3 (C) 1 2 (D) 1 ln x if 0 < x < 3 , (2x − 3) ln 3 if x ≥ 3 then lim f (x) = x→3 (A) ln 3 (B) 3 (C) ln 9 (D) The limit does not exist. PAGE tan(2x) 121 10. lim = x→0 3x (A) 1 3 PAGE 118 11. lim x 3 sin 1 x→0 x 1 (B) 2 = (C) 2 3 (D) 2 (A) −1 (B) 0 (C) 1 (D) The limit does not exist. 1.5 Infinite Limits; Limits at Infinity; Asymptotes OBJECTIVES When you finish this section, you should be able to: 1 Investigate infinite limits (p. 128) 2 Find the vertical asymptotes of a graph (p. 131) 3 Investigate limits at infinity (p. 131) 4 Find the horizontal asymptotes of a graph (p. 137) 5 Find the asymptotes of the graph of a rational function (p. 138) We have described lim f (x) = L by saying if a function f is defined everywhere in x→c an open interval containing c, except possibly at c, then the value f (x) can be made as close as we please to L by choosing numbers x sufficiently close to c. Here c and L are real numbers. In this section, we extend the language of limits to allow c to be ∞ or −∞ (limits at infinity) and to allow L to be ∞ or −∞ (infinite limits). These limits are useful for locating asymptotes that aid in graphing some functions. We begin with infinite limits. Answers to AP® Practice Problems 1. C 2. D 3. D 4. D 5. C 6. C 7. C 8. A 9. D 10. C 11. B AP® Exam Tip Limits at infinity, as well as vertical and horizontal asymptotes, are concepts that commonly appear on the multiplechoice portion of the exam. Teaching Tip Students may be familiar with finding horizontal and vertical asymptotes from prior math classes. In this section, they will learn the formal definition of a horizontal and vertical asymptote and how to use limits to find all horizontal and vertical asymptotes of a function. TRM Alternate Examples Section 1.5 You can find the Alternate Examples for this section in PDF format in the Teacher’s Resource Materials. TRM AP® Calc Skill Builders Section 1.5 You can find the AP ® Calc Skill Builders for this section in PDF format in the Teacher’s Resource Materials. Section 1.5 • Infinite Limits; Limits at Infinity; Asymptotes 127 TE_Sullivan_Chapter01_PART II.indd 10 11/01/17 9:55 am
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