ISM_T11_C02_B - Section 2.5 Infinite Limits and Vertical...

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Unformatted text preview: Section 2.5 Infinite Limits and Vertical Asymptotes c # 93 (d) lim x Ä c1 # lim so the function has no limit as x Ä 0. $Î" $Î% $Î" c $Î% $Î" b (c) xÄ" lim ’ x" c 1 (x c 1) “ œ c_ (d) xÄ" lim ’ x" c 1 (x c 1) $Î% $Î" c $Î% $Î" b 26. (a) xÄ! lim ’ x" c 1 (x c 1) “œ_ (b) xÄ! lim ’ x" c 1 (x c 1) $Î# $Î# c $Î# $Î# b (c) xÄ" lim ’ x" b 2 (x c 1) “œ_ (d) xÄ" lim ’ x" b 2 (x c 1) $Î# $Î# c $Î# $Î# b 25. (a) xÄ! lim ’ x" b 2 (x c 1) “œ_ (b) xÄ! lim ’ x" b &Î$ c &Î$ b 24. (a) tÄ! lim < t " b 7‘ œ _ (b) tÄ! lim c $Î" b 23. (a) tÄ! lim <2 c c and b (e) xÄ! lim xc" x(x b #) œ c_ xc" x(x b #) negative Š positive†positive ‹ negative Š negative†positive ‹ xÄ! lim œ_ 3 t ‘ œ c_ b b $ b (d) xÄ" lim x c 3x b 2 x c 4x # œ lim xÄ" (x c 2)(x c ") x(x c #)(x b 2) œ lim xÄ" c c $ c (c) xÄ0 lim x c 3x b 2 x c 4x # œ lim xÄ! (x c 2)(x c ") x(x c #)(x b 2) œ lim xÄ! (x c 1) x(x b #) (x c 1) x(x b #) b b $ b (b) x Ä c# lim x c 3x b 2 x c 4x # œ x Ä c# lim (x c 2)(x c ") x(x c #)(x b 2) œ x Ä c# b b $ b 22. (a) xÄ# lim x c 3x b 2 x c 4x # œ lim xÄ# (x c 2)(x c ") x(x c #)(x b 2) # # $ (e) xÄ! lim x c 3x b 2 x c 2x # œ œ c_ œ lim (x c 1) x(x b #) xÄ# # # # $ (d) xÄ# lim x c 3x b 2 x c 2x # œ c 2)(x lim (x x (x c c 1) 2) xÄ# c 2)(x lim (x x (x c c 1) 2) xÄ! œ lim xÄ# xc1 x œ lim # c # c # $ c (c) xÄ# lim x c 3x b 2 x c 2x # œ lim xÄ# (x c 2)(x c 1) x (x c 2) œ lim xÄ# xc1 x # b # b # $ b (b) xÄ# lim x c 3x b 2 x c 2x # œ lim xÄ# (x c 2)(x c 1) x (x c 2) # b # $ b 21. (a) xÄ! lim # c (d) xÄ! lim x c1 2x b 4 # œ c" 4 x c 3x b 2 x c 2x œ lim xÄ! b b (c) xÄ" lim x c1 2x b 4 # œ lim xÄ" (x b 1)(x c 1) 2x b 4 œ 2†0 #b4 œ0 (x c 2)(x c 1) x (x c 2) œ c_ œ lim xÄ# xc1 x œ œ " 4 " 4 " 4 ,xÁ2 ,xÁ2 ,xÁ2 negative Š negative††negative ‹ positive " #(4) " 8 negative Š negative†positive ‹ negative Š negative†positive ‹ œ œ (x c 1) x(x b #) œ_ œ_ œ 0 (1)(3) (b) tÄ! lim c b 20. (a) x Ä c# (b) x Ä c# lim œ0 <2 c < " t # # # x c " x œ " # " c ˆ c1 ‰ œ x c1 2x b 4 œ_ $Î" (c) lim # x Ä È2 $ $Î# # x c " x œ 2 # c # " c c (b) xÄ! lim # x # c " x œ 0 b lim xÄ! b b 19. (a) xÄ! lim # c x # (d) x Ä c" lim x x c1 œ x Ä c" lim x (xb1)(xc1) œ c_ negative Š negative†negative ‹ c " x œ 0 b lim xÄ! " cx " cx œ c_ œ_ " Š negative ‹ " Š positive ‹ œ 2c"Î$ c 2c"Î$ œ 0 3 # Š positive ‹ positive x c1 2x b 4 œ c_ positive Š negative ‹ negative Š negative††negative ‹ positive 3 t ‘œ_ b 7‘ œ c_ 2 (x c 1) “œ_ “œ_ “ œ c_ “ œ c_ 94 27. y œ Chapter 2 Limits and Continuity " xc1 28. y œ " xb1 29. y œ " #x b 4 30. y œ c3 xc3 31. y œ xb3 xb2 œ1b " xb# 32. y œ 2x xb1 œ#c 2 xb1 # # 33. y œ x xc" œxb1b " xc" 34. y œ x b" xc1 œxb"b # xc1 Section 2.5 Infinite Limits and Vertical Asymptotes # 95 35. y œ x c% xc" œxb"c $ xc" 36. y œ x2 c " #x b % " œ #x c " b $ #x b % 39. Here is one possibility. 40. Here is one possibility. 41. Here is one possibility. 42. Here is one possibility. # # $ # 37. y œ x c1 x œxc " x 38. y œ x b1 x œxb " x 96 Chapter 2 Limits and Continuity 44. Here is one possibility. 43. 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  $ Ê " lx l " " € B € ! Í lxl  B . Choose $ œ B . Then !  kx c 0k  $ Ê lxl  " B 51. (a) We say that f(x) approaches infinity as x approaches x! from the left, and write lim f(x) œ _, if for every positive number B, there exists a corresponding number $ € 0 such that for all x, x! c $  x  x! Ê f(x) € B. (b) We say that f(x) approaches minus infinity as x approaches x! from the right, and write lim f(x) œ 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 $ Ê f(x)  cB. ! b xÄx ! c # # Ê kx b 5k  # Now, 1 (x b 5) € B € ! Í (x b 5)  " ÈB # " B Í kx b 5k  x Ä c& (x b 5) " " ÈB . Choose $ œ " ÈB . Then 0  kx c (c5)k  $ Ê " (x b 5) € B so that lim œ _. xÄx # 50. For every real number B € 0, we must find a $ € 0 such that for all x, 0  kx c (c5)k  $ Ê # # 2 $ œ É B , then 0  kx c 3k  $ Ê # # Now, 2 (x c 3) c2 (x c 3)  cB  0 so that lim # c2 (x c 3)  cB  ! Í €B€0 Í (x c 3) 2  " B Í (x c 3)#  x Ä $ (x c 3) c2 2 B 2 Í !  kB c $k  É B . Choose œ 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  $ Ê # # Ê c" x "  cB so that lim c x œ c_. xÄ! # # c " x  cB  ! Í " x €B€0 Í x  # " B Í kxk  " ÈB . Choose $ œ " ÈB , then 0  kxk  $ Ê kxk  " lx l € B. Now, " Ê " lx l € B so that lim x Ä ! 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  $ Ê c" x  cB. Now, " ÈB œ _.  cB. € B. Section 2.5 Infinite Limits and Vertical Asymptotes (c) We say that f(x) approaches minus infinity as x approaches x! from the left, and write lim f(x) œ 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! Ê f(x)  cB. 52. For B € 0, 53. For B € 0, " x " " € B € 0 Í x  B . Choose $ œ B . Then !  x  $ Ê 0  x  " B 97 54. For B € !, 55. For B € 0, Ê " xc# " xc# " " € B Í !  x c 2  B . Choose $ œ B . Then #  x  # b $ Ê !  x c #  $ Ê !  x c 2  $ Ê # 57. y œ sec x b " x 58. y œ sec x c # c # " #B . " 1cx Then " c $  x  " Ê c$  x c 1  0 Ê " c x  $  € B for !  x  1 and x near 1 Ê xÄ" # 56. For B € 0 and !  x  1, " 1cx b € B € ! so that lim xÄ# " xc# œ _. " B " Í (" c x)(" b x)  B . Now " "cx " #B € B Í 1 c x#  lim  1 since x  1. Choose " " Ê (" c x)(" b x)  B ˆ 1 b x ‰  B # 1bx # œ _. " x c " 2 c $  x  2 Ê c$  x c 2  ! Ê c B  x c 2  0 Ê " xc# "  cB Í c x c # € B Í c(x c 2)  c " Ê cB x Ê " x "  cB  0 Í c x € B € 0 Í cx  " x  cB so that lim xÄ! " x " B " " Í c B  x. Choose $ œ B . Then c$  x  ! œ c_. " B " " " Í x c 2 € c B Í x € 2 c B . Choose $ œ B . Then " xc#  cB  0 so that lim xÄ# " xc# œ c_. " B b Ê " x € B so that lim ! c xÄx xÄ! " x œ _. 98 Chapter 2 Limits and Continuity # 59. y œ tan x b " x 60. y œ " x c tan x 2.6 CONTINUITY 1. No, discontinuous at x œ 2, not defined at x œ 2 2. No, discontinuous at x œ 3, " œ lim g(x) Á g(3) œ 1.5 c xÄ$ # $Î" 63. y œ x#Î$ b " x 64. y œ sin ˆ x # # 61. y œ x È4 c x 62. y œ c" È4 c x 1‰ b1 ...
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