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ma1012_hw8_201012 - Section 2.4 One-Sided Limits and Limits...

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Unformatted text preview: Section 2.4 One-Sided Limits and Limits at Infinity 87 Mathematics I: Homework 8. One sided limits and limits at infinity. xÄ1 xÄ1 (c) 1th dition, œ 1 since 2, right-hand : 1 3, 7, 11, Thomas’ Calculus 1Yes,Elim1 f(x) Chapter the Section 4and, left-hand 13, 21, 25. Ä 86 Chapter 2 Limitsxand Continuity (b) lim f(x) œ " œ lim f(x) Thomas’ Calculus 11th Edition, Chapter 2, Section 5: 1, 7, 11, 17, 21, 23, 25. end do: Mathematica (assigned Total: 23+18=41 function and values for x0, eps and del may vary): Clear[f, x] y1: œ L c eps; y2: œ L b eps; x0 œ 1; f[x_]: œ (3x2 c (7x bf(x) œ 0 œ 5)/(x c 1) (b) lim 1)Sqrt[x] b lim f(x) xÄ1 Plot[f[x], {x, x0 c 0.2, x0 b 0.2}]x Ä 1 (c) Yes, lim f(x) œ 0 since the right-hand and left-hand L: œ Limit[f[x], x ÄÄ 1 x x0] limits eps œ 0.1; del œ 0.2; exist and equal 0 Plot[{f[x], y1, y2},{x, x0 c del, x0 b del}, PlotRange Ä {L c 2eps, L b 2eps}] limits exist and equal 1 TIP Find . 2.4 ONE-SIDED LIMITS AND LIMITS AT INFINITY Answer: 1. (a) True (e) True (i) False ŸxŸ2 2. (a) True y Ÿ 1 and y œ 2 (e) True xists for c belonging to (i) True #) Chapter 2.4 (b) True (f) True (j) False (b) False (f) True (j) False 2 # (c) False (g) False (k) True (c) False (g) True (k) True (d) True (h) False (l) False (d) True (h) True 3. (a) xÄ# lim f(x) œ xÄ# b " œ #, lim f(x) œ \$ c # œ " xÄ# xÄ# 4 # (b) No, lim f(x) does not exist because lim f(x) Á lim f(x) xÄ# (c) xÄ% lim f(x) œ 4 # b 1 œ 3, lim f(x) œ xÄ% b"œ\$ r (c"ß ") r ("ß _) 82. (d) Yes, lim f(x) œ 3 because 3 œ lim f(x) œ lim f(x) _  92 _ Chapter 2 Limits and % x x Ä Continuity x Ä % Section 2.4 One-Sided Limits and Limits at Infinity xÄ% ŸyŸ1 3b 4 2 3x . (a) x 4 œ lim f(x) œ œ œ 1, 3 b 4t œ 3 ,œ ˆt c # ‰ ", f(2) œ 2 œ " œ lim f(x) xists for c belonging to7b 4(aÑ 81. lim lim lim lim f(x) \$ œ " œ (b) x lim f(x) 5# Ä1 x Ä „ _ 2x c 5 x x Ä„ _ 2 c x Ä# 87 "Îx ( lim ˆ " ‰ œc) limlimz œf(x) œ 3 cx(c1) œ 4, z 1, ˆz œ " ‰ xÄ_ x (c) lim f(x) (b) Yes, lim f(x) œ 1 because " œ lim f(x) œ Yes, xlim1 f(x) œ 1 since the right-hand and left-hand Ä xÄ# xÄ# xÄ# xÄ z Ä ! c" x Ä c" Ä# t Ä 0 x 2 c 5t # x xÄ1 lim f(x) œ 3 c (c1) œ 4 x Ä c" limits exist and equal 1 b 4h b 5b2.5 INFINITE LIMITSb 4) c5 h(h AND VERTICAL ASYMPTOTES 2 œ lim œ È0 b 4 œ È5 5 h b 4h b 5 b È5‹ h Ä ! h ŠÈh b 4h b 5 b È5‹ negativeb È5 2x 9. (a) domain: 0 Ÿ x Ÿ 2 3x 5. lim œ c_ 6. lim œ_ Š positive ‹ b x Ä c& 2x5 10 x Ä c) x b 8 " range: 0  y ŸŠ positive œ 2 1 and y ‹ 1. lim 3x œ _ 2. lim 2x œ c_ positive xÄ! xÄ! (b) xlimc f(x) exists for c belonging to Ä positive 4 c" 7. lim (xc7) œ _ (0ß 1) r ("ß #)Š positive ‹ 8. lim x (xb1) œ c_ xÄ! xÄ( positive 3 " c) x 3. lim xc2 œ(c_ œ 2 4. lim xc3 œ _ Š negative ‹ xÄ# Ÿ g(x) Ÿ Èx when x  0 positive Š negative ‹ 25. (b) No, 1 limsing(x) does not exist since Èx is not defined for x  0 2 " 3 " " 84. x lim ˆ x c cos x ‰ ˆx Ä ! x ‰ œ lim a3)# c cos )b (1 b sin )) œ (0 c 1)(1 b 0) œ c1, ˆ) œ x ‰ b Ä_ Ä ) È ! exist since lim g(x) does not exist È È bpositive È (c) No, lim g(x) does not b 5 c È5 4h b 5 " 3.œ lim xc2 hœbc_5 c 5 ‹ Š Èh Š negativeb È5 ‹ 4. ‹ Š positive ‹ xÄ! x Ä ! lim lim Š3 4hhb x c3 œ _ positive h b 4h b 5 b 5 "‰ 83. lim ˆ3 b 2 ‰ ˆcos " ‰ œ lim (3 b 2))(cos )) œ (3)(1) œ 3, ˆ) œ È 6. (a)x Yes, xlim " )positive 0 by the sandwich theorem since c x x g(x) œ Ä " 5 x b 6 ‰ ˆ 3 c x ‰ x Ä „ _ ˆ 1 b 6 ‰ ˆ 3 c 1 ‰ Ä ˆ Š‰ ˆ 7 0 ˆ 2 ‰ " œ_ x œ ! positive ‹ œ 1 ‰7 1. 7 lim ˆ 13x 1 ‰ 1 2. lim 2x œ c_ œb x 7 # 1 xÄ! xÄ! " ˆ3 b 2 ‰ ˆcos " ‰ œ lim (3 b 2))(cos )) œ (3)(1) œ 3, ˆ) œ x ‰ x x xÄ „_ )Ä0 4 3b x 3 b 4t 3 " 81. lim 53x b 4 œNo,lim . (a) x Ä „ _ 2 c 5 doesÄ 0 exist since, sintˆ " ‰xdoes not approach any single value as x approaches 0 lim f(x) œ t lim 2 c 5t œ # ˆ œ ‰ not 2x c 5 xÄ „_ x 2 2 xc" 1c1 x È x È 8 (a) 3 (b) f(x) œ É c0.5 blim É33/2 . cos " ‰ ˆ1 bÄ ! "1‰ œ lim É )# c œ É b b # b sin ))œlim c 1)(10 œ0) lim cf(x) ˆ) œ " ‰ 1 œ 84. 0.5 b 1 œ ˆ 1/2 œ c c sin 2. lim a3 x b # cos ) " (1 œ 0 x Ä(0 œ! 1 b x Ä 1 1, œ x x Ä _ x 11. (b)x lim f(x)x œ x)Ä 1!0 œ 0 lim Ä (c) Yes, lim f(x) œ 0 since the right-hand and left-hand xÄ! z xÄ! " "Îx "‰ xÄ1 82. x lim ˆ x (c) lim f(x)œ 1, notœ x ‰ because lim f(x) does not exist œ lim z does ˆz exist Ä_ limits exist and equal 0 2(c2) z Ä ! b5 c2 " 13. bbc52) AND VERTICAL 1( xÄ ˆ 2x b x ‰ œ ˆ c# b " ‰ Š (LIMITSx‹Ä ! (2) ˆ # ‰ œ 1 ASYMPTOTES! œ 2.5 INFINITE c#) x 92 83. lim d) Yes, lim f(x) œ 4 because 4 œ lim f(x) œ lim f(x) Chapter (2 Limits Ä c" Continuity and x x Ä c" 21. 1 xÄ# hÄ! xÄ\$ Chapter 2.5 1. 7. negative Š negative ‹ positive Š negative ‹ 9. (a) 5. 10. (a) x Ä c) lim xÄ! lim xÄ! lim 2 3x 2x x b8 2 10. x (d) x œ 0 xÄ\$ [email protected] Š negative ‹ œ c_ positive œ domain: c_  x  _ (a) _ range: c" Ÿ y Ÿ positive 1 œ_ (b) (b) 6. xÄ! lim 2 3x negative Š positive†positive ‹ positive Š positive ‹ œ c_ 3x 2xb10 xÄ! lim x Ä c& x lim œ_ 2 œ c_ c" negative Š negative ‹ negative 4 (x 3x ) (x c 1) 3x negative negative x c 3x b 2 " (x negative negative #c c ! 2x ! 2x lim x(xc 2)(x c "2)positive lim x(x_#) positive œ" x. œ1# c1xlimxx#tan#4x œlimœlim c 2)(x!c2)(x #b"2)blimœ x(xc 1)(x œ1))bœx_ #_limx8 lim xcœ œ _ x ÄŠ negative†positive ‹ ‹ ‹ 13. _ lim c Ä2! xc 4x _ _ (x c c œ lim ‹ œ (c)(a) lim.# 13x limxœ 2 3œ œ c(x x Ä c ")c œ ) Š positivelim x(xc sec 14. (4)4.œ Ä sec x " 3 _ lim 3xc # bc 3x b Š negative†positive )(x 14. Š positive x positive Ä x ! ÄÄ ˆ xxÄ x b Ä Ä ‰x c c c 2 0 # 1 Ä 0x(1xb cscÄ _ xc_x Äx(x c##)(x b#)(x x Änegative! b##) ‰b # lim csc )ˆ) œ4x)) œ x Ä ! x ! x(x 2) Ä x Ä ˆ x‰ \$ Ä! (x c c (x 2)(x c " (x (x c 0 1) negative x b lim b (d) limlimx bccœc23xœ 2 lim clim c 2)(x3b"2) limc)lim c 1)lim 1)) 0x(x(1)(3) œ _ positive positive lim x c 3x4x 23x x c3 positive (x c2)(x c ") x(xc ) œpositivec x #(x œ œ positiveœ 0 œ œ0 (b) ‹ " " Š positive positive œ _ #)(x x(x x(x ) b xœ . œpositive ‹ _ Š limÄ " b c Š positive ‹ x Änegative " c _Ä_ Ä3"x c Šxpositivex‹4x "x Ä.‹ x(lim(c #)(xb)cx#)(xxnegative ‹xc_b##(1)(3)negative 4. Š negativeŠ c3 ‹ 4. x lim xc3 Šnegative†positive œ "x x Ä c#limŠ4x xŠÄ œ 3 x(x Äb) #b 2)xc2 limcx2)1 x(x1 Ä1cœ c_b #) ‹ x lim negative œ _ Ä \$ ( Š positive ‹negative ‹ cpositive " Ä # b) lim œ 2x _ œ2x œ4c_ Š 2 ‹ cx x positive positive 2x b# b4 b 4 2x x Ä # positive negative Ä\$ 3x Ä x c " negative cx c negative ) lim. œ ) cscc and c scc )cot )x _ cx_ and__cc(2 c cot ))ÄŠcÄ), œ: lim so thedoes not exist sided limits and limits anegative ‹ 15. lim) cotxlim œ))2 œ _ x(2 c #œŠ)#positive_, 5 " bc ‹ Š t infinity. negative c Homework _ 6. x lim c 1)xc 1) 2†) † †œ ) Šso the positive 1) 2 l(x b)œ blim(1Äb"#c_x0b82lim _limathematics positive†limit c‹ œ 8. One Ä c& 2xb10 œ _ lim (2 x(xcb#1)(x)cœ3x2bcœ0 limM (x c 2)(xcot_œI†positive ‹ (x limit does not exist (e)(c) (x 1)(x ) œ 1) im Ä Š negative†positive ‹ 2) !)! Ä x1)(x c 0 œ ) lim!limÄ 4!(xb 2x b #xb0c#4x 4) œ4xœ 0 ! x(x c #)(x b positivex Ä ! x(x b #) x Ä ! x Äb x(x œ 4 œ 0Ä 0 Ä Ä ! b# Ä x Ä "b 2x 4 x4 c 3x b2x b " 2x negative (x c 2)(x c ")negative negative (x c 1) negative 2x 3x 3x negative negative œ œ c_ ‹ ‹ lim œ œ_ Š œ 0‹ Š negative ‹ Š negative ‹ (d) lim lim " x œ2 œ c_ lim x c c " œ lim œ and Š " c" and5. lim x #) 4x b_ 5._ lim c #)(xŠŠ positivelim †positive ‹) positive 6. œ 0 c& 2xb106. _ lim negative† ) xÄ Ä lim lim x c" c ot )) œxc_Äx(xc)c 4b #œ c_x"Ä)c) limb8bŠ c cotpositive _,not the (1)(3) does not exist œ c_ c& 2xb10 negative ‹ 16. x Ä. !x"x"andb x(x)) 8"œ x Ä"and)x(x _, (22)positive Ä)"œ x(x b # exist " xlim lim (2Ä cot x )(2 c cot œ x so the negativedoes so limit limit Ä ! " Š positive†positive m 4 lim47!œxxcÄ œ) (xc7) 2)(xc_2)(xc2) )œ _ ) x" lim Ä !11. 2) œ _ Ä ! lim Š8. x Ä ! x ‹ positive ‹ negative Š positive†positive †positive(xb1) )Ä 4x b positive ‹ so (e) c# lim x Ähasxnoblimitlimit as x 0. 0. so # the function ( c " Äœ no(xas x Ä Ä x Ä the function (x # c_ # has Š positive†positive ‹ x(x b #) positive negative negative 4 4 c" c" " x Ä ! lim "lim " " Š positive ‹ Š œ c_ ‹ Š positive†positive ‹ m limc 2)(x clim 1)" c 7) 2)(x27_2)(xc2) œ c7)_ Š positive ‹negative†negative †negative lim x (x2b1) œ c_ ) x (x 47.(x x "2)(xœ " limcœ (x.b œ c_", c œ _ œ (x cc (x (x1) " " ‹ 8. lim 1) 2)(x 17. negativeŠ8. 2 , cim ccÄ ( x c_ œ c2)_, Ä ( (x 4 # c b# _ positive " œ Ä l# " negative Špositive † positive ‹! Ä im xim andc<2)Ä‘2) lim œ œlim x_ Š positiveŠ negative†negative ‹x Š ‘ Š negative††negative (b) ‘ lim # (a)x9(x lim lim 3 Äxc c x3x œ œ œc x 17.l œlimimxx(x c bxc 2)x(x b #)_œ _ b2)(xc2) œ _ Š negative(b)positive‹ c‹negative†negative"positive ‹ _ Ä ! x (xb1) positive†positive l .c(a)(x (x2)(xc2) 3x _ (x Š†positive‹2 ‹ 3 _œ 3x negative c! (a) Ä x x 2x # 2 cxœc‘ _ cÄœ lx4 ! Ä ! Ä x Ä !Ä 4 œ x # im <Ä # t c (b) positive lim <t2 cxœpositive_ lim positive 3 tÄ ! † †positive †< ! " " " t Ä !t Ä ! Š" t)Ä ! (x c (x œt(x ccœ Ä" c 2)(x 2)(x im œ t lim!2)(x c 1)4lim c 1)lim 2)(xc2)œ"x c"1 "œ x" Áx2Á 2 1) Š" ‹ 2 x c (xb2)(x c2) œ c_ 1x cœ" c_ 1 b †negative ‹ positive " l Ä c# the x Ä c#œ has#no "2 im#ximxlsoimx 2)functionÄ(xc2lim limit4as x4 Á 2 ,20. _ im x x (x " xlim lim œ" , 2 c œb)xc(x c(a)c 2) lim œœ 9.lim _ œ x ,Ä c œ _ (x Ä 4 limc bxcx2) 2)x œ x x_ ,(xbxlim , 4 , l# cœ# x , Š positivepositive(b) Š†negative†negativec_lim ‹ lim 9. # (x (x 2)(x4Ä #3x ÄÄ(a) 2)(xc2) œ œ‹ (b) œ c_ c# Ä #x Ä Ä # 4 c c 21. # # positive 2 †negative x Ä 3x 3x x x" Ä! x Ä ! 3x Ä! xÄ! " 2)(x(a) cc 1) c 1) 2 x œ1x c "x c 1 " " x lim " " x lim " 10. b 7 b c (x <2)(x 2)(x 2 c _œ _ < (b)7‘xœ†‹ œ "‹ œ < negative negative ‘ imim lim 4 œx(xc‘xœ7_œœ 2)(xc2)œ1 "c2) œ x" Áx2Á 2 (b) (b) Šlim" Š†negative7c_xc_ c_ )l "œ (xl<c" cc 1)4lim ‘ "lim limb2)(xœ x œ _ (a) l x ct# t œÄ œ (xc _ lim " t Ä im#ximlim(x lxlimxÄ c#xlimœ # œ (x x 4x, 4 Á 2 , œ c_ Š Ä !t Ä ! b‹ b !3 im lim , " "4 ! xb t positivenegative t < Š positive†negative ‹ 3 t.cc!Ä Älim 2)xc 2)cx 2) Äc2) xc_ _ (xb2)(xc2) ÄÄ(œx c (x (x cb2)(x# œ xlim Ä # 4 t c) # Äx # xx(a)x !# im <2 (xx c4‘ Ä # Äc (b) †negative 2 c t ‘ œ _ lim xxc 1) c# c 1) 23. x Ä c Ä cc 1) Ä# # t 2 2 tÄ! c 2)(x ! c 2)(x (x c 2 "c t10. œ (b) lim x 2 œ (b)_ lim x 2 œ c_ c lim ximÄ 2)(x (x lœ limxlimœ1_ 1 "œ x" Áx2Á 2 _ œ(x l (x c (x(a) lim œx x 10.c (a), x Á 2 , im c 1x x œ lim , x œ Ä! x Ä! ! x Ä x(œ11.(xlim cx2) 4"#"œx# Äœ _ 4x Ä 4 " x œ _ # " # d) # 2)im2) x Ä x Äœ lim x4x 4 œ _ Ä xlc lim 2 2)(xc xlim 12." x Š2negative† œ lim Š negative"negative ‹ positive x " "negative ‹ x Ä"! œ _ positive lim x x " (xbc4 2 2) # ax b(xb2)(xc2) x4 † " clim 2 mlimlim’cÄ1)b #œcx1)lim1)(xc_œ _ œ _ ) c(xxclimxcÄcÄ1) b (xb“ œx1) !œ # 1) xœ x c ! x # Ä Šlim " Š†positive †‹ ! “ œ _œÄ ! ax b x (xb1)(xc Ä c_ x bpositive ‹ “ x _ Ä (b) negative†negative ’positive c 1) (b) positive b lim 1 (x x ’ 1 2)(x †negative ’ negative 2(a) x Ä 2)(x 2)(x c (x cx Äc 1) “ , x Ä †negative x ‘ œ x c" xx (x Ä bc25. œ 1) " œ Š negativeŠ negative††negative<(x"c 1)(x 7‘ œ c_ Š positive!x Ä ! x"Ä !ximc 2) c 2) cc_ 7_ _ .!lim! Ä lim x x< " œ (xœ c _ (a)l (x ! (b) x b positive†negative ‹ ‹ ‹ positive lim t x Ä x Ä ! (x x (x t 2) 4 4 4 4 " " " ! t11. ! x " 2 œx 11. xlim Ä" lim x œ œ _ Š t Ä Š " positive x " ‹œ 12. lim xœ _ lim 12. 2 lim œ œ_ 2 lim œ c_xœ _ lim mlimlim13. xxcÄ œb lim1)(xc_œ Ä1) b œ c_x Ä ! ax (d) (d) positive positive xlim! positive œÄ !x Ä !b ) x c’ ’ lim x (xb“ œœ _ c b (c) x Älimxœ xblimcx1)Äc 1)x (xb!x a_! x lim limx " 14. (x c†1)(x c21) secxx _ _ “ ’ †b b‹ “ œ _ œ ax ’positive negative “ ! x (x " x Ä 1)(x x Ä x Ä ! ax b 1 1) positive negative lim x x "Ä œ Äl" lim1 (x x tan œ lim " im x " Ä 18. x _ (xb1)(xc1) œ _ Š positiveÄ " ‹ Š positive†positive ‹ x Ä " †positive x xc1 "(a) (xˆ x ‰c1 1) b1)(xc œ xÄˆ ‰ x Ä Ä Ä) (x c x " x 2)(x 2)(x " x Ä " c 1) c 1) "1) " 2)(x "" (x (x (x (x 2 " negative negative x ") 2 imœ x(xxlimcc#ccc#"2) cœ)(xœ (x(x x(xxbx(xc(4) œ "œ " " im (a)limc1#)(xxx(x 1lim blimb2 xx(x“b#œ_x#œ ) #_8#(4) 8œ 8 ) limxlim œ b)(x b)(x b 2) lim limœ œ 1)bœ œ lim 1)(xc1) ) _ Š lim† positive ‹ 2) x .c# x Ä Ä"# c xlimœ " tanxc13. Äb#1)(xc) tan(4) œ _ (b) positive14. ’ x "†‹b positive 14. _ lim sec x œ _ positive Š negative negativec 1) “ œ _ xx"Ä x c x(x x # xcim! ’c c x Ä # Äc_ lim# # œ c_ x_ # x (x x 1) Ä x lim (x sec x œ (b)13." œ c"œ Š positive†negative ‹" Š1positive†negative ‹ xÄ! (xˆ 1)(xc1) c1 œ x lÄ lim (xˆ x ‰c1 1)1 œ lim b‰ " b1)(xc 1 " 1 Ä " Äc )x Ä ˆ “‰œ “ œ (x " (x negative x l im 15. "x2)(x c" cc bcœ2Äœ x Ä 1) (x 1) (b) (b) limx ’ x (x c 1)(x cŠnegative‹ c‹ Ä ˆ ‰ lim ’ c x c Šnegative2 c_ _ ’ lx Äcc 2)(x 2)(x œ “ " œ lim (x x(a) b2 c lim c) (x c ") 1) œimx limx c’(x)(x(x(x2)1)b"2)“csc _œ _ c x(xcx1) cœ œ " xÄ! Ä! x x lx œ Ä !im #x cx(x (1#)(x 1) 2) )) “ œ _ œ )_ œ 4xÄ ! limlim)x(x’ !#)(x bx b lim lim x(x_#) bx(x b #)_ _ # negative† †positive negative (d) lim Š negative(x negative†positive ‹ bnegativepositive œ Ä(c)# x Ä lc#Ä x c x (x c 1)œ _# c# x Ä c# im b c(œ x(x c Ä clim #c b c)x Ä " lim (xxbc1 xœ x Ä x Ä (xb1)(xc1) œ _ Š positive†negative ‹"’ xŠ1positive†c 1) “‹ _ xÄ" x c1 1 " 1 negative xx"Äc"" c ") c "“cœxc_1) c 1) c 1) Äcc""c 2)(x 1)(x1 1) “(x c c_ c)c c Ä c(x (x " c (c) lim(xim(x ’2)(x œ1) œ) limx2)(x (x œ (d) (d) c Š c Šnegative “ œ c‹ œ negative ’ “ negative‹ _ 2l x c 1) imimxx(x’cx(x cx(x (xb)(x(xb 1)œ )lim#)cœ )(1 #)_ _) œ c_ x limx limx ’ x (x c 1)(xnegativec_positive ‹ œ 15. Ä" Ä" x xœ" xÄ " #)(x b)(x c2)xbc x lim Ä ! bx(x_b œ ) Ä !l15. lim # (1 lim Ä x ) blim _ b csc # 2) c 2) csc x(x œ negativeŠ negative† †positive †positive ! Ä Ä ! ) lim (2 c cot )) ) x(x !_ and lim (2 c cot )) œ _, so the limit does not exist Ä! ! œc # Ä 16. Ä ! c 2)(x 2)(x! " ") c ") (x’ c (x (x 0 0 .2 (a) (xim(x cc "cx2)(xclimc11) (x “ 1) c 1) cœ œÄ ! 0œ 0 lim) Ä) c œ(x œ limœ _ 1) )0 0œ (b) lim ’ x " c (x c11) “ œ c_ imœ x(x cx(x cx(x cb)(x b 2) lim bx(x bx(x(1)(3)(1)(3)(1)(3) lim l œ œ œ x(x xÄ! x Ä x ÄÄ )(x b)(x # 2)x Ä x Ä x Ä "#) #) b #) " " x "# ! # 2) "" c 16. lim " (2 c cot )negative _(2 c cot )) œ c_ and œ _, so the limit œ _, so exist 16. ) œnegativeand lim (2 c cot )) lim (2 c cot )) does1not the limit does not exist lim 1 negative Ä ! _ )Ä (d) ! lim ’ x " c (x c 1) “ œ c_ ) “ lim œ (c) 17. ) Ä ! lim p(xŠ "1)†Šœ †œ ‹†positive ‹ "Ä ! œ _ _ _ lim(a)’ x c ositive positive c ‹ ) c Š cpositive positive Š positive"positive ‹ xÄ" xÄ" x c4 positive (xb2)(xc2) † "x c " œ x(x b #)_17. #)_ œ _ " negative Š Šnegative negative ‹"" ‹ " † negative positive " " (b) lim negativeŠœ †positive (xb2)(xc2) œ lim negative lim (a) x lim x 17.positive †# (xb2)(xc2) œ c_ (xb2)(xc2) œ _ _ c4 (a) Ä lim Ä # x c4 œ x limÄ # x c4 œ x Ä # xÄ# xÄ# x slhaslimitxas xasÄ Ä 0. " " no no limit 0. 0. imit as Ä x "" " " (c) lim lim lim œ (b) x Ä c# x x 4 4 (œ limlim#(xb2)(xc2) c2) lim c_b2)(xc2) œ c_ (xb2)(x c c œb) x Ä c x c4 œœ c_ (x xÄ# xÄ# " Š positive"negative‹ Š positive†positive ‹ † " " Š positive†negative ‹ Š positive†negative ‹ Š positive"positive ‹ † ‘c_ _ œ c_ (d) (c) (d) 7‘_ _ œ 18. (a) 18. (b) (a) 2 2 “ œ1) œ“_ _ “_ œ 1) c (x (c) (b) 2 2 “ œ1) œ“_ _ “_ œ 1) c (x (c) 1 1 “ œ1) œ“_ “_ œ 1) c (x xÄ# lim x Ä c# x Ä c# lim " " " (b) lim b) < lim2 3 <‘ œ ‘ 3 ‘ œ _ "c3 " lim ( œ c) tlim! (xx "< œ 2 c _ lim x c4 (œ x (b)c# lim2 c4 tc2) tœ _ _ b2)(xc2) t x c4 œ ÄlimÄ (xÄ2)(xc2) œ c_ (x Ät ! b2)(x t bc ! xÄ# # xÄ xÄ# x x Ä c# Ä c# x Ä c# œ c_ œ_ " " Š negative"negative‹ Š Š positive††negative ‹ positive†negative ‹ " Š positive†negative ‹ lim (œ lim d) lim œ œlim _ x x Ä c# x c4 (b) (b)cc# <xlim4 " 7bœbœ‘c_ 2)(xc2) x x lim lim x" 2)(xc2) " 7 c_ (xb Ä # (xb c b < < lim x c1 œ limÄ(b)b1)(xc1) œ‘x Ä‘c7 œ c_ _# t ÄtÄtÄ xÄ" x Ät" !(x !t ! t x lim xÄ" Ä" " "" " Š negative"negative ‹ Š negative"negative ‹ † † positive Š positive†positive ‹ positive positive Š positive†negative‹ Š positive†positive ‹ negative positive Š positive†negative ‹ Š positive†negative ‹ negative Š positive†negative ‹ positive Š positive†positive ‹ positive Š positive†negative ‹ negative Š positive†negative ‹ x x x x x x lim (xb1)(xc1) œ lim (xb1)(xc1) œ _ x 18. œ x lim (xb1)(xc1) œ c_ x c1 œ lim x c1 œ _ c1 (a) Ä " xÄ" " xÄ " "x Ä 2 "" 2 2 (b) (b) (b) ’ lim b ’ xc 1)b “ œ1) œ“_ _ lim lim ’ b (x (x “_ œ Ä !Ä ! lim x x œœ xlimx Ä (xxx 1)(xx œ(xœ_ c 1) c x lim ! xxb x c _ (x 1 cc x lim " x x 1 1 (b) limÄ "(xb1)(xc1) c1) lim 2 b1)(xc1) œ c_ Äc x x " c" x c" œ"x Ä 2 Ä xÄ" xÄ "" 2 (d) (d) (d) ’ lim b ’ xc 1)b “ œ1) œ“_ _ lim lim x ’ b “_ œ (x x x x Ä x Ä x Ä x xœ (xœ (x c 1) c " " x" lim c) lim lim (xx 1)(xc1) lim _ (xb1)(xc1) œ _ x c1 (œ b c1 lim x Ä c" x x Ä c" Ä c" x Ä c" _ (b) (b) (b) ’ lim c ’ xc11)c 1“ œ1) œ“c_ _ lim lim x " ’ x " (x " (x c 1) c1“c_ œ c c (x x Ä x Äx Ä ! !! x Ä x Äx Ä " "" 1 1 “ œ1) œ“c_ _ “c_ œ c 1) c (x (d) (d) (d) ’ lim c ’ xc11)c 1“ œ1) œ“c_ _ lim lim x " ’ x " (x " (x c 1) c1“c_ œ c c (x [email protected] ...
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