100000000 100000 1000 100 10 10 100 1000 100000

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Unformatted text preview: .1 0.25 0.5359 0.5132 0.5001 0.5000 0.5000 0.4999 0.4881 0.4721 0.6 The limit is 1/2. -0.25 0.25 0 45 Chapter 2 (b) 0.25 0.1 0.001 0.0001 8.4721 20.488 2000.5 20001 100 The limit is +∞. 0 0.25 0 (c) −0.25 −0.1 −0.001 −0.0001 −7.4641 −19.487 −1999.5 −20000 0 -0.25 0 The limit is −∞. -100 19. (a) −0.25 −0.1 −0.001 −0.0001 0.0001 0.001 0.1 0.25 2.7266 2.9552 3.0000 3.0000 3.0000 3.0000 2.9552 2.7266 3 The limit is 3. -0.25 0.25 2 (b) 0 −0.5 −0.9 −0.99 −0.999 −1.5 −1.1 −1.01 −1.001 1 1.7552 6.2161 54.87 541.1 −0.1415 −4.536 −53.19 −539.5 60 The limit does not exist. -1.5 0 -60 Exercise Set 2.1 20. (a) 46 0 −0.5 −0.9 −0.99 −0.999 −1.5 −1.1 −1.01 −1.001 1.5574 1.0926 1.0033 1.0000 1.0000 1.0926 1.0033 1.0000 1.0000 1.5 The limit is 1. -1.5 (b) 1 0 −0.25 −0.1 −0.001 −0.0001 0.0001 0.001 0.1 0.25 1.9794 2.4132 2.5000 2.5000 2.5000 2.5000 2.4132 1.9794 2.5 The limit is 5/2. -0.25 0.25 2 21. (a) −100,000,000 −100,000 −1000 −100 −10 10 100 1000 2.0000 2.0001 2.0050 2.0521 2.8333 1.6429 1.9519 1.9950 100,000 100,000,000 2.0000 2.0000 40 asymptote y = 2 as x → ±∞ -14 6 -40 (b) −100,000,000 −100,000 −1000 −100 −10 10 100 1000 20.0855 20.0864 20.1763 21.0294 35.4013 13.7858 19.2186 19.9955 100,000 100,000,000 20.0846 20.0855 70 asymptote y = 20.086. -160 160 0 47 Chapter 2 −100,000,000 −100,000 −1000 −100 −10 10 100 1000 100,000 100,000,000 −100,000,001 −100,000 −1001 −101.0 −11.2 9.2 99.0 999.0 99,999 99,999,999 (c) 50 no horizontal asymptote -20 20 –50 22. −100,000,000 −100,000 −1000 −100 −10 10 100 1000 100,000 100,000,000 0.2000 0.2000 0.2000 0.2000 0.1976 0.1976 0.2000 0.2000 0.2000 0.2000 (a) 0.2 -10 10 asymptote y = 1/5 as x → ±∞ -1.2 −100,000,000 −100,000 −1000 −100 −10 10 100 0.0000 0.0000 0.0000 0.0000 0.0016 1668.0 2.09 × 1018 (b) 1000 1.77 × 10301 100,000 100,000,000 ? ? 10 asymptote y = 0 as x → −∞, none as x → +∞ -6 (c) 0 6 −100,000,000 −100,000 −1000 −100 −10 10 100 0.0000 0.0000 0.0008 −0.0051 −0.0544 −0.0544 −0.0051 1000 100,000 100,000,000 0.0008 0.0000 0.0000 1.1 asymptote y = 0 as x → ±∞ -30 30 -0.3 23. (a) lim x→0+ sin x x (b) lim x→0+ x−1 x+1 (c) lim (1 + 2x)1/x x→0− Exercise Set 2.2 24. (a) 25. lim+ (a) x→0 48 cos x x (b) lim+ x→0 1 x+1 (c) (b) y lim− 1 + x→0 2 x x yes; for example f (x) = (sin x)/x x 26. no (b) yes; tan x and sec x at x = nπ + π/2, and cot x and csc x at x = nπ , n = 0, ±1, ±2, . . . (a) The plot over the interval [−a, a] becomes subject to catastrophic subtraction if a is small enough (the size depending on the machine). (c) 29. (a) It does not. EXERCISE SET 2.2 1. (a) −6 (b) 13 (c) −8 (d) 16 (e) 2 (f ) −1/2 (g) The limit doesn’t exist because the denominator tends to zero but the numerator doesn’t. (h) (a) 0 (b) The limit doesn’t exist because lim f doesn’t exist and lim g does. (c) 0 (f ) 2. The limit doesn’t exist because the denominator tends to zero but the numerator doesn’t. The limit doesn’t exist because the denominator tends to zero but the numerator doesn’t. (d) 3 (g) The limit doesn’t exist because (e) 0 f (x) is not defined for 0 ≤ x < 2. (h) 1 3. (a) 7 (b) −3 (c) π (d) −6 (e) 36 (f ) −∞ 4. (a) 1 (b) −1 (c) 1 (d) −1 (e) 1 (f ) −1 5. 0 6. 3/4 10. 12 11. 15. 0 20. 3 −3 9. 0 13. 3/2 14. 4/3 0 18. 19. √ −5 24. √ −1/ 6 8 −4/5 12. 16. 3/2 7. 0 17. 21. √ 1/ 6 22. √ √ 8. 5 23. 5/3 √ 3 4 27. does not exist 28. −∞ 29. −∞ +∞ 32. does not exist 33. does not exist 34. −∞ 36. −∞ 37. −∞ 38. does not exist 39. −1/7 41. 6 42. +∞ 43. 44. 4 25. +∞ 26. 30. +∞ 31. 35. +∞ 40. +∞ 3 +∞ 49 Chapter 2 45. +∞ +∞ 49. (a) 2 (b) 2 (c) 2 50. (a) −2 (b) 0 (c) does not exist 51. (a) 3 46. 47. −∞ 48. +∞ y (b) 4 1 x 52. (a) −6 53. (a) Theorem 2.2.2(a) doesn’t apply; moreover one cannot add/subtract infinities. (b) lim+ 54. 56. 57. 58. 59. 60. 61. (b) F (x) = x − 3 x→0 1 1 −2 xx x→0− lim 1 1 + x x2 lim √ x→0 x = lim+ = lim− x→0 x→0 x−1 x2 = −∞ x+1 = +∞ x2 55. lim x→0 x √ 1 x = 4 x+4+2 x2 =0 x+4+2 √ x2 + 3 + x 3 √ lim ( + 3 − x) = lim √ =0 2+3+x 2+3+x x→+∞ x→+∞ x x √ x2 − 3x + x −3x 2 − 3x − x) √ lim ( x = lim √ = −3/2 2 − 3x + x 2 − 3x + x x→+∞ x→+∞ x x √ x2 + ax + x ax x2 + ax − x √ = lim √ = a/2 lim x→+∞ x2 + ax + x x→+∞ x2 + ax + x √ √ a−b x2 + ax + x2 + bx (a − b)x 2 + ax − 2 + bx √ √ √ x x = lim √ = lim 2 + ax + 2 + bx 2 + ax + 2 + bx x→+∞ x→+∞ 2 x x x x x2 lim p(x) = (−1)n ∞ and lim p(x) = +∞ x→+∞ x→−∞ 62. If m > n the limits are both zero. If m = n the limits are both 1. If n > m the limits are (−1)n+m ∞ and +∞, respectively. 63. If m > n the limits are both zero. If m = n the limits are both equal to am , the leading coefficient...
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