thomasET_226348_ism12

thomasET_226348_ism12 - 84 24. (a) Chapter 2 Limits and...

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Unformatted text preview: 84 24. (a) Chapter 2 Limits and Continuity &$ 27. y oe " xc1 28. y oe " xb1 29. y oe " #x b 4 30. y oe c3 xc3 31. y oe xb3 xb2 oe1b " xb# 32. y oe 2x xb1 oe#c 2 xb1 $% $" c $% $" b (c) x" lim ' x" c 1 (x c 1) " oe c_ (d) x" lim ' x" c 1 (x c 1) $% $" c $% $" b 26. (a) x! lim ' x" c 1 (x c 1) "oe_ (b) x! lim ' x" c 1 (x c 1) $# $# c $# $# b (c) x" lim ' x" b 2 (x c 1) "oe_ (d) x" lim ' x" b 2 (x c 1) $# $# c $# $# b 25. (a) x! lim ' x" b 2 (x c 1) "oe_ (b) x! lim c &$ b t! lim < t " b 7` oe _ (b) t! lim < " t b 7` oe c_ 2 (x c 1) ' x" b "oe_ "oe_ " oe c_ " oe c_ Section 2.5 Infinite Limits and Vertical Asymptotes # # 85 33. y oe x xc" oexb1b " xc" 34. y oe x b" xc1 oexb"b # xc1 39. Here is one possibility. 40. Here is one possibility. # # $ # 37. y oe x c1 x # 35. y oe x c% xc" oexb"c $ xc" 36. y oe x2 c " #x b % " oe #x c " b $ #x b % oexc " x 38. y oe x b1 x oexb " x 86 Chapter 2 Limits and Continuity 42. Here is one possibility. 41. Here is one possibility. 43. Here is one possibility. 44. Here is one possibility. 45. Here is one possibility. 46. Here is one possibility. 48. For every real number B 0, we must find a $ 0 such that for all x, ! kx c 0k $ " lx l B ! lxl " B. Choose $ oe # # kx b 5k # Now, 1 (x b 5) B ! (x b 5) " B # " B kx b 5k x c& (x b 5) " " B . Choose $ oe " B . Then 0 kx c (c5)k $ " (x b 5) B so that lim oe _. # 50. For every real number B 0, we must find a $ 0 such that for all x, 0 kx c (c5)k $ # # 2 $ oe B , then 0 kx c 3k $ # # Now, 2 (x c 3) c2 (x c 3) cB 0 so that lim # c2 (x c 3) cB ! B0 (x c 3) 2 " B (x c 3)# x $ (x c 3) c2 2 B 2 ! kB c $k B . Choose oe c_. 1 (x b 5) # 49. For every real number cB 0, we must find a $ 0 such that for all x, 0 kx c 3k $ # # c" x " cB so that lim c x oe c_. x! # # c " x cB ! " x B0 x # " B kxk " B . Choose $ oe " B , then 0 kxk $ kxk " lx l B. Now, " " B. Then ! kx c 0k $ lxl " B " lx l B so that lim x ! lx l c2 (x c 3) # 47. For every real number cB 0, we must find a $ 0 such that for all x, 0 kx c 0k $ c" x cB. Now, " B oe _. cB. B. Section 2.5 Infinite Limits and Vertical Asymptotes 51. (a) We say that f(x) approaches infinity as x approaches x! from the left, and write lim f(x) oe _, if for every positive number B, there exists a corresponding number $ 0 such that for all x, x! c $ x x! f(x) B. (b) We say that f(x) approaches minus infinity as x approaches x! from the right, and write lim f(x) oe c_, if for every positive number B (or negative number cB) there exists a corresponding number $ 0 such that for all x, x! x x! b $ f(x) cB. (c) We say that f(x) approaches minus infinity as x approaches x! from the left, and write lim f(x) oe c_, if for every positive number B (or negative number cB) there exists a corresponding number $ 0 such that for all x, x! c $ x x! f(x) cB. 52. For B 0, 53. For B 0, " x " " B 0 x B . Choose $ oe B . Then ! x $ 0 x " B 87 54. For B !, 55. For B 0, " xc# " xc# " " B ! x c 2 B . Choose $ oe B . Then # x # b $ ! x c # $ ! x c 2 $ # 57. y oe sec x b " x 58. y oe sec x c # c # " #B . " 1cx Then " c $ x " c$ x c 1 0 " c x $ B for ! x 1 and x near 1 x" # 56. For B 0 and ! x 1, " 1cx b B ! so that lim x# " xc# oe _. " B " (" c x)(" b x) B . Now " "cx " #B B 1 c x# lim 1 since x 1. Choose " " (" c x)(" b x) B ^ 1 b x B # 1bx # oe _. " x c " 2 c $ x 2 c$ x c 2 ! c B x c 2 0 " xc# " cB c x c # B c(x c 2) c " cB x " x " cB 0 c x B 0 cx " x cB so that lim x! " x " B " " c B x. Choose $ oe B . Then c$ x ! oe c_. " B " " " x c 2 c B x 2 c B . Choose $ oe B . Then " xc# cB 0 so that lim x# " xc# oe c_. " B b " x B so that lim ! c xx ! b ! c xx xx x! " x oe _. ...
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This note was uploaded on 04/17/2008 for the course MA 113 taught by Professor Massman during the Spring '08 term at Rose-Hulman.

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