mathematical constant

68 5. Chapter 2  •  Limits Use the graph of ƒ in the figure to find the following values or state that they do not exist. T a. ƒ112  b. lim ƒ1×2  c. ƒ102  d. lim ƒ1×2 xS1 xS0 y 4 3 y 5 f (x) T 1 21 2 3 4 x 21 t – 9 . 1t – 3 a. Make two tables, one showing values of g for t = 8.9, 8.99, and 8.999 and one showing values of g for t = 9.1, 9.01, and 9.001. t – 9 b. Make a conjecture about the value of lim . t S 9 1t – 3 10. Let ƒ1×2 = 11 + x2 1>x. xS0 c. What mathematical constant does lim 11 + x2 1>x appear to xS0 equal? 22 6. Let g1t2 = a. Make two tables, one showing values of ƒ for x = 0.01, 0.001, 0.0001, and 0.00001 and one showing values of ƒ for x = -0.01, -0.001, -0.0001, and – 0.00001. Round your answers to five digits. b. Estimate the value of lim 11 + x2 1>x. 1 22 9. 11. Explain the meaning of lim+ ƒ1×2 = L. xSa Use the graph of ƒ in the figure to find the following values or state that they do not exist. 12. Explain the meaning of lim- ƒ1×2 = L. xSa a. ƒ122  b. lim ƒ1×2  c. lim ƒ1×2  d. lim ƒ1×2 xS2 xS4 13. If lim- ƒ1×2 = L and lim+ ƒ1×2 = M, where L and M are finite xS5 xSa y xSa real numbers, then how are L and M related if lim ƒ1×2 exists? xSa 6 3 T 5 y 5 f (x) 4 14. Let g1x2 = x – 4x . 80x – 20 a. Calculate g1x2 for each value of x in the following table. b. Make a conjecture about the values of lim- g1x2, lim+ g1x2, xS2 3 xS2 and lim g1x2 or state that they do not exist. xS2 2 1 0 T 7. Let ƒ1×2 = 1 2 3 4 5 6 x 1.9 1.99 1.999 1.9999 x3 − 4x g 1x 2 = 8∣ x − 2 ∣ 2.1 2.01 2.001 2.0001 x x3 − 4x g 1x 2 = 8∣ x − 2 ∣ x2 – 4 . x – 2 a. Calculate ƒ1×2 for each value of x in the following table. 2 b. Make a conjecture about the value of lim xS2 x – 4 . x – 2 x 1.9 1.99 1.999 1.9999 x2 − 4 ƒ 1x 2 = x − 2 2.1 2.01 2.001 2.0001 x x 15. Use the graph of ƒ in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. a. ƒ112 b. lim- ƒ1×2 c. lim+ ƒ1×2 d. lim ƒ1×2 xS1 xS1 xS1 y 2 x2 − 4 ƒ 1x 2 = x − 2 y 5 f (x) 1 3 T 8. Let ƒ1×2 = x – 1 . x – 1 a. Calculate ƒ1×2 for each value of x in the following table. x3 – 1 b. Make a conjecture about the value of lim . xS1 x – 1 x ƒ 1x 2 = 0.9 0.99 0.999 0.9999 1.1 1.01 1.001 1.0001 0 1 2 x 16. What are the potential problems of using a graphing utility to estimate lim ƒ1×2? xSa 3 x − 1 x − 1 x x3 − 1 ƒ 1x 2 = x − 1 M02_BRIG3644_03_SE_C02_056-130.indd 68 23/06/17 9:54 AM 2.2 Definitions of Limits 17. Finding limits from a graph Use the graph of ƒ in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. a. ƒ112 b. lim- ƒ1×2 c. lim+ ƒ1×2 d. lim ƒ1×2 e. ƒ132 f. lim- ƒ1×2 g. lim+ ƒ1×2 h. lim ƒ1×2 i. ƒ122 j. k. lim+ ƒ1×2 l. lim ƒ1×2 xS1 xS1 xS3 xS1 xS2 T xS3 xS3 lim- ƒ1×2 23. ƒ1×2 = xS2 xS2 24. ƒ1×2 = T 5 y 5 f (x) 26. ƒ1×2 = 1 – x4 ;a = 1 x2 – 1 27–32. Estimating limits graphically and numerically Use a graph of ƒ to estimate lim ƒ1×2 or to show that the limit does not exist. EvalxSa 4 uate ƒ1×2 near x = a to support your conjecture. 3 27. ƒ1×2 = 28. ƒ1×2 = 1 1 21 2 3 4 5 x 18. One-sided and two-sided limits Use the graph of g in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. a. g122 b. lim- g1x2 c. lim+ g1x2 d. lim g1x2 e. g132 f. lim- g1x2 xS2 g. lim+ g1x2 xS2 xS3 h. g142 xS3 x – 100 ; a = 100 1x – 10 x2 + x – 2 ;a = 1 x – 1 2 xS2 x 2 – 25 ;a = 5 x – 5 25. ƒ1×2 = y 69 i. lim g1x2 xS4 y 29. ƒ1×2 = 30. ƒ1×2 = 31. ƒ1×2 = 32. ƒ1×2 = x – 2 ;a = 2 ln 0 x – 2 0 e2x – 2x – 1 ;a = 0 x2 1 – cos12x – 22 1x – 12 2 ;a = 1 3 sin x – 2 cos x + 2 ;a = 0 x sin1x + 12 0x + 10 ; a = -1 x 3 – 4x 2 + 3x ;a = 3 0x – 30 33. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. 5 x2 – 9 does not exist. xS3 x – 3 b. The value of lim ƒ1×2 is always found by computing ƒ1a2. a. The value of lim 4 y 5 g(x) xSa 3 c. The value of lim ƒ1×2 does not exist if ƒ1a2 is undefined. 2 d. lim 1x = 0. (Hint: Graph y = 1x.) xSa xS0 e. 1 0 1 2 3 4 5 x lim cot x = 0. (Hint: Graph y = cot x.) x S p>2 34. The Heaviside function The Heaviside function is used in engineering applications to model flipping a switch. It is defined as H1x2 = e Practice Exercises 19–26. Evaluating limits graphically Sketch a graph of ƒ and use it to make a conjecture about the values of ƒ1a2, lim- ƒ1×2, lim+ ƒ1×2, and lim ƒ1×2 or state that they do not exist. xSa xSa 19. ƒ1×2 = b x2 + 1 3 20. ƒ1×2 = b 3 – x x – 1 if x 6 2 ;a = 2 if x 7 2 1x 21. ƒ1×2 = • 3 x + 1 if x 6 4 if x = 4; a = 4 if x 7 4 if x … – 1 ; a = -1 if x 7 – 1 xSa 0 1 if x 6 0 if x Ú 0. a. Sketch a graph of H on the interval 3- 1, 14. b. Does lim H1x2 exist? xS0 35. Postage rates Assume postage for sending a first-class letter in the United States is $0.47 for the first ounce (up to and including 1 oz) plus $0.21 for each additional ounce (up to and including each additional ounce). a. Graph the function p = ƒ1w2 that gives the postage p for sending a letter that weighs w ounces, for 0 6 w … 3.5. b. Evaluate lim ƒ1w2. w S 2.3 c. Does lim ƒ1w2 exist? Explain. wS3 22. ƒ1×2 =  x + 2  + 2; a = -2 M02_BRIG3644_03_SE_C02_056-130.indd 69 17/07/17 11:16 AM 80 Chapter 2  •  Limits 15. Suppose g1x2 = b if x ≠ 0 if x = 0. 2x + 1 5 hS0 xS0 16. Suppose 4 x + 2 if x … 3 if x 7 3. xS3 does not exist. xS3 17. Suppose p and q are polynomials. If lim xS0 p1x2 q1x2 110x – 9 – 1 x – 1 53. lim 15 + h2 2 – 25 = 10 and 54. xS4 lim 5x xS – 9 xS1 27. lim pS2 29. lim xS3 5x 2 + 6x + 1 8x – 4 3p 14p + 1 – 1 – 5x 14x – 3 xS6 lim 1t 2 + 5t + 72 tS – 2 3 2 26. lim 2 t – 10 tS3 28. lim 1x 2 – x2 5 xS2 30. lim hS0 31. lim 15x – 62 3>2 32. lim 33. lim 34. lim xS2 xS1 x2 – 1 x – 1 x 2 – 16 35. lim xS4 4 – x 37. lim xSb 39. 1x – b2 50 – x + b x – b lim xS – 1 41. lim xS9 x + 1 1x – 3 x – 9 43. lim a tS5 12x – 12 2 – 9 hS0 xS3 lim xSa M02_BRIG3644_03_SE_C02_056-130.indd 80 x 2 – 2x – 3 x – 3 1x + b2 7 + 1x + b2 10 41x + b2 1 1 5 + h 5 40. lim hS0 h 42. lim a wS1 1 1 b 61t – 52 t 2 – 4t – 5 x – a ,a 7 0 1x – 1a 100 110h – 12 11 + 2 xS – b 2 44. lim a a4t b16 + t – t 22b tS3 t – 3 45. lim 3 116 + 3h + 4 3t 2 – 7t + 2 36. lim tS2 2 – t 38. xS1 31x – 421x + 5 3 – 1x + 5 x 58. lim ,c ≠ 0 S x 0 1cx + 1 – 1 59. lim x cos x xS0 xS1 22. lim 4 xS1 56. lim x – 1 14x + 5 – 3 xS4 20. lim 1-2x + 52 23. lim 12x 3 – 3x 2 + 4x + 52 24. 25. lim x – 1 1x – 1 xS1 57. lim 19–70. Evaluating limits Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers. 21. w + 5kw + 4k 2 ,k ≠ 0 wS – k w2 + kw xS2 Practice Exercises 19. lim 13x – 72 1 2 b – 2 x – 2 x – 2x lim 55. lim xS2 xS2 h hS0 18. Suppose lim ƒ1×2 = lim h1x2 = 5. Find lim g1x2, where xS2 52. lim a 2 q102 = 2, find p102. ƒ1×2 … g1x2 … h1x2, for all x. xSc x 2 – 2cx + c2 x – c 1 1 15 x 2 + 2x 50. lim xS3 x – 3 51. lim xS1 Compute lim- ƒ1×2 and lim+ ƒ1×2. Then explain why lim ƒ1×2 xS3 48. lim 1 1 x 4 49. lim xS4 x – 4 Compute g102 and lim g1x2. ƒ1×2 = b 116 + h – 4 h 47. lim 46. lim xSa 1 1 b w – 1 w – w 61. lim xS0 63. xS0 65. lim+ tS2 sin 2x sin x 62. lim cos x – 1 cos2 x – 1 xS0 1 – cos x cos2 x – 3 cos x + 2 lim- 60. lim xS0 x2 – x 0x0 64. lim w S 3- 0 2t – 4 0 t2 – 4 x2 – 1 66. lim g1x2, where g1x2 = c x + 1 xS – 1 -2 67. lim xS3 69. lim- xS1 0w – 30 w2 – 7w + 12 x – 3 0x – 30 68. lim xS5 x3 + 1 1x – 1 70. lim if x 6 – 1 if x Ú – 1 0x – 50 x 2 – 25 x – 1 x S 1+ 2x 2 – 1 71. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers. a. If lim ƒ1×2 = L, then ƒ1a2 = L. xSa b. If lim- ƒ1×2 = L, then lim+ ƒ1×2 = L. xSa xSa c. If lim ƒ1×2 = L and lim g1x2 = L, then ƒ1a2 = g1a2. xSa d. The limit lim xSa 2 ƒ1×2 g1x2 xSa does not exist if g1a2 = 0. e. If lim+ 2ƒ1×2 = 2 lim+ ƒ1×2, it follows that xS1 xS1 lim 2ƒ1×2 = 2 lim ƒ1×2. xS1 xS1 x 2 – a2 , a 7 0 1x – 1a 28/06/17 9:22 AM
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