41. lim x - 1 42. lim 1- Vx 1 1 1-x 71. f(x) = x2 +1, ifx so 1-x, ifx > 0 43. lim *+x2-273 x - x2 44. lim x+x2- 2x3 x - x2 72. f(x) = 3x + 2, ifx 1 57. lim sin x 58. lim cotx x+ 1, ifx 3 NI 60. lim - 1 3 tan 1 x 77. f(x) = sin x, if x I sin x 1 0 sec-1 x lim (3+ h)2. - 32 2x - 1, ifx 50 63. 64. lim (2+ h)2 - 22 h- 0 78. f(x) = 47 -1 h - 0 2 x - 1' -, ifx > 0 65. lim (1 + h)3 -13 66. lim (-1 + h)3 - (-1)3 h Use the Squeeze Theorem to find each of the limits in Exer- 1 cises 79-86. Explain exactly how the Squeeze Theorem applies 67. lim 1+h 68. lim 2+h 2 in each case h- h h 79. lim x sin = 80. lim x sin 1 * - 0 1 - 0 69. lim (2+ h)2 70. lim (1 + h)2 81. lim(ex - 1) sin 82. lim sin x sin = h- x-+0 Describe the intervals on which each function f in Exer- 83. lim(x - 1) cos _ 1 84: lim(x - 1) cos cises 71-78 is continuous. At each point where f fails to be continuous, use limits to determine the type of discontinuity 85. lim xtan 1 X 86. lim x2 tan -1 1 and any left- or right-continuity. Applications In Exercise 88 in Section 0.1, you constructed a piecewise- (a) Calculate the value of T(63,550) and the limit of T(m) defined function from the 2000 Federal Tax Rate Schedule as m approaches 63,550 from the left and from the that you will use in the next two problems. Specifically, you right. found that a person who makes m dollars a year will pay T(m) (b) Use part (a) to argue that the function T(m) is dollars in tax, given by the function continuous at m = 63,550. What does this mean in real-world terms? 0.15m, if 0 5 m 288,350. (b) Use part (a) to argue that the function T(m) is contin- 87. Suppose you make $63,550 a year and pay taxes according uous at m = 288,350. What does this mean in real- to the given formula. world terms?pints by evaluation? 12. Explain how the algebraic function (a) Make a table of values that describes the behavior of the quantity (1 + h)1/hash -> 0. f (x) _ (9- + 1)(4 - 3x) (b) Make a table of values that describes the behavior of 3x2 the quantity eh - 1 is a combination of identity, constant, and power func- as h -> 0. tions. Why does this mean that we can calculate limits of (c) What do your tables of values have to do with Defi- this function at domain points by evaluation? nition 1.25 and Theorem 1.26? Skills Calculate the limits in Exercises 25-28, using only the conti- nuity of linear and power functions and the limit rules. Cite 31. lim 3x 32. lim (3.1x2 - 4x + 0.8) each limit rule that you apply. X 1 2 x-1.7 4 + 2x 25. lim 15(3 - 2x) 26. lim x- 1 33. lim 34. lim 4+2x X - 2 X 1 2 x -> - 1 ( x + 4) (x + 2) x- -2 x+2 x2- 1 x2 - 1 3 35. lim 36. lim 27. lim (3x + x2(2x + 1)) 28. X 1 x 1 r-0x-1 -0 2x2 - 4x + 1 xx + 3 3x2 + 8x - 3 37. lim X- 3 3x2 1 8x 3 38. lim Calculate each of the limits in Exercises 29-70. * - - 3 x+3 29. lim (x2 - 1) 30. lim(x - 1) (x + 1)(x + 5) 4x - 2 6x 2+ x - 2 39. lim 40. lim X-0 x- 2 X- 1/2 6x2 +x-2 x- -2/3 3x + 2 136 Chapter 1 Limits 1 - Vx 42. lim 71. f(x) = Ja +1, ifxso 41. lim 1 -x 1 -x ifx - 0 x- 14 2 23- 2x3 DA