数学分析讲义 第四版 (刘玉琏 傅沛仁 著) 高等教育出版社 课后答案 第二单元.pdf

8 111 1 SSS1 1 1 SSS1 2 5 SSS1 3 9 111 444 13 SSS2 1 13 SSS2 2 21 SSS2 3 36 SSS2 4 40 SSS2 5 51 111nnn YYY 57 SSS3 1 57 SSS3 2 64 111ooo YYY555 75 SSS4 1 75 SSS4 2 83 111 95 SSS5 1 95 SSS5 2 100 SSS5 3 109 SSS5 4 113 SSS5 5 114 111888 nnn444 AAA 125 SSS6 1 125 SSS6 2 136 SSS6 3 142 SSS6 4 148 111 170 SSS7 1 170 SSS7 2 172 SSS7 3 179 SSS7 4 187 111lll 194 SSS8 2 194 SSS8 3 204 SSS8 4 210 SSS8 5 232 111 244 SSS9 1 244 SSS9 1 252 SSS9 1 nnn 266 SSS9 2 273 SSS9 2 290 SSS9 3 298 SSS9 4 309 1 111 323 SSS10 1 323 SSS10 2 132 SSS10 3 334 SSS10 4 352 111 366 SSS11 1 366 SSS11 2 375 SSS11 3 378 SSS11 4 389 111 222 CCC 393 SSS12 1 393 SSS12 2 400 SSS12 3 406 111 nnn 425 SSS13 1 425 SSS13 1 430 SSS13 2 493 111 ooo 465 2 1 4 S K2 1 5 6 1 45 1 eA Q 4 lim n an a d nd 1 0 N N n Nk an a d 0 N N n Nk an a N N n N an a 0 N N n Nk an a 1 1 0 N N n N an a 0 N1 N n N1 an a N1 Kk 0 N N n N N1 an a 2 k N N N n Nk k an a 0 0 1 k N 1 k 1 1 k l 1 k 0 k 0 k N k an a 0 k an a 0 z N N n N k an a 00 du 00 0 U y 0 X n 1 1 n n N n 1 1 n U u 3 4 lim n an a d 4 0 k an k an a d an k an a N k an a N k an a 0 3 a2k 1 k N ka2k 1 1 1 0 1 n 34 u 4 4 lim n an a d 5 k N kk an u m a 1 k a 1 k d fi k N an a 1 k a 1 k an a Nk k an a 1 k 1 k 0 k d 2 u 5 4 lim n an a d 3 3 y e 4 2 lim n cosn n 0 y cosn n 0 cosn n 1 n l 1 n 1 N h 1 i u 0 N h 1 i N n N cosn n 0 7 5n2 7n n2 5 35 7 n 35 n 7 35 7 N max n 7 h35 7 io u 0 N max n 7 h35 7 io N n N k 5n2 7n n2 5 0 n 1 n 0 n 1 n n 1 n n 1 n 1 n 1 n 1 2 n l 1 2 n 1 4 2 N h 1 4 2 i u 0 N h 1 4 2 i N n N k n 1 n 0 0 n 4 n3 2n 0 n3 2n n3 n n 1 n 2 n 3 4 24n2 n 1 n 2 n 3 24 n 1 1 2 n 1 3 n 24 n 1 1 2 1 4 192 n 1 l 192 n 1 192 1 N max n 4 h192 1 io u 0 N max n 4 h192 1 io N n N k n3 2n 0 0 N N n N k an l k 3 a n 0 3 a n 3 p an 0fi lim n an a 0 a 0 N1 N n N1 k an a 0 q 0 N N N N1 n N k an a 0 l k 3 a n 3 a an a 3 a n 2 3 a n 3 a 3 a 2 an a 3 a2 1 3 a2 lim n 3 a n 3 a 3 ea 0fi lim n an a 0 N1 N n N1 k an a a an 0 N N N N1 n N k an a an 0 l k 3 a n 3 a an a 3 a n 2 3 a n 3 a 3 a 2 an a 3 a2 0 n 2 2n n 0 2n n 2 1 2 2 2 3 2 n 1 2 n 2 2 n 4 n l 4 n 4 N max n 2 h 4 io u 0 N max n 2 h 4 io N n N k 2n n 0 lim n 2n n 0 2 lim n n n 1 y fi n 1 k n n 1 a n 0 n 1 an n d n n N n 2 k 1 an n 1 nan n n 1 2 a2 n a n n n n 1 2 a2 n 6 a2 n 2 n 1 l n 2 k 0 n n 1 a n 2 n 1 2 n 1 n 2 0 n n 1 2 n 1 l 2 n 1 4 2 1 N max n 2 h 4 2 1 io u 0 N max n 2 h 4 2 1 io N n N k n n 1 O nn 1 n 2 n 1 n 1 n n n 2 n n n 1 n n 1 n g 0 1 n n 0 1 n l 1 n 1 2 N h 1 2 i u 0 N h 1 2 i N n N k 1 n n 0 0 1 1 2 1 2 3 1 n n 1 1 1 1 n 1 1 1 n 1 1 1 N h1 1 i u 0 N h1 1 i N n N k 1 1 2 1 2 3 1 n n 1 1 0 N N n0 N k n0 n0 1 0 n0 n0 1 1 2 lim n n n 1 6 0 10 y 2 1 n u y a 2 0 1 N N n 2k N k N k 2 1 2k a 1 a 0 1 a N k N k 2 1 2k 1 a 3 a 0 1 u a R k lim n 2 1 n 6 a 2 1 n u S K2 2 5 6 1 64 2 y e lim n an a K lim n an a K 1 n y fi lim n an a N N k an a l k an a an a 0 N N k bn b 0 N N k bn M u 0 N N k b2 n b 2 bn b bn b bn b bn b M b M b lim n b2 n b2 6 4 y e lim n an a N kan 0 y fi lim n an a 0 N N n N k an a a l an a a 0 7 y e an 1 q an 0 q 1 n 1 2 3 K lim n an 0 1 1 1 ey o 4 lim n an 0 e 8 y a16 0 Ka1 0 Ka2 0 k8B a1 a2 an 0 an z 0 w k lim n an 0 a1 6 0 fi n 1 k a2 q a1 n 2 k a3 q a2 q2 a1 J 8B y k Nk ak 1 q ak qk a1 fi lim k qk 0 0 q 0 K N k K k qk 0 qk l ak 1 0 qk a1 0 lim n n a n r 0 r q 1 k lim n n a n r Nk n a n q l 0 an qn fi lim n qn 0 0 q Y n k lim n an 0 9 y ean 0 lim n an 1 an r 0 r q 1 k lim n an 1 an r Nk an 1 an q l an 1 qan fi n N 1 kaN 2 qaN 1 n N 2 kaN 3 qaN 2 q2aN 1 J 8B y k NkaN k 1 qaN k qkaN 1 fi lim k qk 0 0 q 0 K N k K qk 0 qk l aN k 1 0 qkaN 1 aN 1 aN 1 lim n an 0 10 an a1 a2 1 an 1 an an 1 n 2 3 4 JA 8B y an 1 5 n 1 5 2 n 1 5 2 no y an an 1 5 1 2 0 618 y n 1 ka1 1 5 n 1 5 2 1 5 2 o 1 n 2 ka2 1 5 n 1 5 2 2 1 5 2 2o 1 n 3 k a3 a1 a2 1 1 2 1 5 n 1 5 2 3 1 5 2 3o 9 n k 1 n k k ak 1 1 5 n 1 5 2 k 1 1 5 2 k 1o ak 1 5 n 1 5 2 k 1 5 2 ko Kak 1 ak ak 1 1 5 n 1 5 2 k 1 5 2 ko 1 5 n 1 5 2 k 1 1 5 2 k 1o 1 5 n 1 5 2 k 1 1 5 2 1 1 5 2 k 1 1 5 2 1 o 1 5 n 1 5 2 k 1 1 5 2 2 1 5 2 k 1 1 5 2 2o 1 5 n 1 5 2 n 1 5 2 no u an 1 5 n 1 5 2 n 1 5 2 no A 1 5 2 B 1 5 2 kan 1 5 An Bn B A 1 5 1 5 B A 1 A B 1 Kak 1 ak ak 1 lim n an an 1 lim n 1 5 An Bn 1 5 An 1 Bn 1 lim n 1 B A n 1 A h 1 B A ni 1 A B 5 1 2 0 618 11 e 4 5 lim n 1 a a2 an 1 b b2 bn a 1 b 0 1 i k y am max a1 a2 ak k am n an m n pan 1 an 2 an k n pkan m am n k fi lim n n k 1 Y n k lim n n pan 1 an 2 an k am max a1 a2 ak 13 y lim n 1 n2 1 n2 1 n2 n2 n 0 y n N k 1 n n 1 n2 n 1 n2 1 n2 1 1 n2 n Y n k lim n 1 n2 1 n2 1 n2 n 0 14 y ea1 2 a n 1 2a n n 1 2 K an 4 y k 8B y an k 2 fl n 1 a1 2 2 a n 2 K an 1 2a n 2 2 2 11 g an O fl n N k fi n N a n 0 n an lim n an a k lim n a2 n 1 2 lim n an an 1 2a n a2n 1 2an a2 2a a 2 u lim n an 2 15 y ea0 a 0 an 1 2 an 1 2 an 1 n 1 2 K an 4 y dA L n N k an 1 2 an 1 2 an 1 s an 1 2 an 1 2 a2 n 2 an ke q n N k an 1 an 1 2 an 2 an an 2 a2 n 2an 0 an N n an 34 lim n an b k lim n an lim n 1 2 an 1 2 an 1 b 1 2 b 2 b b 2 d4 S5 b 0 u lim n an 2 16 A OKy e 4 5 1 xn a0 a1q anqn 0 q 1 ai M i 0 1 2 y n p N k xn p xn an 1qn 1 an 2qn 2 an pqn p an 1 qn 1 an 2 qn 2 an p qn p M qn 1 qn 2 qn p M 1 q q n 1 qn p 1 0 N N n N kqn 1 l p N k xn p xn M 1 qq n 1 M 1 q 12 M 1 q OK xn 3 xn 1 1 1 1 2 1 n y fi n N k 1 n 1 1 2 3 n 1 1 2 2 2 z n 1 1 2n 1 n p N k xn p xn 1 n 1 1 n 2 1 n p 1 2n 1 2n 1 1 2n p 1 2 12n 1 2n p 1 2n 1 1 2n p 1 0 N N n Nk 1 2n 1 l p N k xn p xn 1 2n 1 0 K N k K k ank a a ank nK n N anK anl ka an n N kan N k a an a lim n an a 18 y ex1 a y1 b xn 1 x nyn yn 1 1 2 xn yn n 1 2 3 K xn yn 34 4 y A O L n N k xn 1 x nyn 1 2 xn yn yn 1 qkxn x nxn x nyn xn 1 yn 1 1 2 xn yn yn 1 1 2 yn yn yn n N kxn yn qyn 1 yn y1 xn b xn NO k b n xn N ke a e n 13 34 lim n xn A lim n yn B l k lim n yn 1 lim n 1 2 xn yn B A B 2 u A B lim n xn lim n yn A B 19 y ean 1 1 2 1 n lnn K an 34 4 c c 0 577216 y ky n N k 1 n 1 ln 1 n 1 1 n 1 fi n N k 1 1 n n e g k nln 1 1 n 1ln 1 1 n 1 1 n2 1 1 1 n n 1 n n n n 1 n n 1 n n 1 1 1 n 1 n 1 1 n n 1 n 1 1 n n 1o qfi lim n 1 1 n n 1 e l n N ke g k 1 n 1 ln 1 1 n 1 n 1 ln 1 1 n gy an y an ke fl n N dfi k an 1 an 1 n 1 ln n 1 lnn 1 n 1 ln 1 1 n 1 ln 1 1 ln 1 1 2 ln 1 1 n lnn ln2 ln 2 3 ln n 1 n lnn ln2 ln3 ln2 ln n 1 lnn lnn ln n 1 lnn lnn ln 1 1 n 0 an ke p an 34 lim n an lim n 1 1 2 1 n lnn c 20 y e 3 c n N k x2 x1 x3 x2 xn xn 1 0 n N n N p N yn p yn xn p xn xn xn p xn xn 1 xn 1 xn 2 xn p 1 xn p xn 1 xn xn 2 xn 1 xn p xn p 1 yn p yn xn 21 y e n N k yn p yn 0 n N n N p N sn p 1 sn 1 cn cn 1 cn p 1 xn p xn xn p xn p 1 xn p 1 xn 1 xn 1 xn xn 1 xn xn 2 xn 1 xn p xn p 1 cn cn 1 cn p 1 xn 15 22 x m sinx 0 1 m V x0 m x1 m sinx0 xn m sinxn 1 K xn 34 y n Nk xn 1 xn m sinxn m sinxn 1 sinxn sinxn 1 2 sin xn xn 1 2 cos xn xn 1 2 2 sin xn xn 1 2 cos xn xn 1 2 2 xn xn 1 2 xn xn 1 n 1 x2 x1 x1 x0 n 1 x3 x2 x2 x1 2 x1 x0 J 8B y k N k xk 1 xk k x1 x0 e x1 x0 Kx0 x1 x2 w xn 34 e x16 x0 k xn p xn xn p xn p 1 xn p 1 xn 1 xn 1 xn xn 1 xn xn 2 xn 1 xn p xn p 1 n n 1 n p 1 x1 x0 x1 x0 1 n fi lim n n 0 0 N n Nk n l k xn p xn x1 x0 1 n 0 m N n m k an a 0 N N N m n N k A n l k 1 n a1 a2 an a a1 a a2 a an a n a1 a a2 a am a n am 1 a an a n 16 A n n m n 0 lim n an a K lim n a 1a2 an a y ea 0 d123K k lim n a1 a2 an n 0 dA O L O n a 1a2 an a1 a2 an n 2 Y n k lim n n a 1a2 an 0 ea 0 k lim n 1 an 1 a d123K k lim n a1 a2 an n a lim n 1 a1 1 a2 1 an n 1 a lim n n 1 a1 1 a2 1 an a qfi N LA A O L O n 1 a1 1 a2 1 an a n a 1a2 an a1 a2 an n Y n k lim n n a 1a2 an a A d J y 1 lim n an 1 an an 0 n 1 2 K lim n n a n a y a0 1 A J k lim n n a n lim n n s an an 1 an 1 an 2 a2 a1 a1 a0 2 lim n n n 1 y fi lim n n n 1 1 n 2 an n n 1 a1 1 A J k lim n n n lim n n s n n 1 n 1 n 2 2 1 1 1 3 lim n 1 n n 0 17 y fi lim n 1 n 0A J k lim n 1 n n lim n n s 1 n 1 n 1 1 2 1 1 0 4 lim n n n n e y fi lim n 1 1 n n lim n n 1 n n e A J k lim n n s 2 1 1 3 2 2 4 3 3 n 1 n n lim n n 1 n n e u lim n n n n lim n n 1 n n 1 n n e 25 y e lim n an a lim n bn b K lim n a1bn a2bn 1 anb1 n ab y bn b n bn b n dfi lim n n 0 u a1bn a2bn 1 anb1 n a1 b n a2 b n 1 an b 1 n b a1 a2 an n a1 n a2 n 1 an 1 n d123K k lim n b a1 a2 an n ab e y lim n a1 n a2 n 1 an 1 n 0 fi lim n an a l an k M 0 n N k an M qfi lim n n 0 0 m N n mk n 0 N N N m n N k A n 0 0 1 2x 0 2x x 0 u 0 A log2 0 x A k 2x 0 0 1 x sin 1 x 0 1 x 1 A 1 u 0 A 1 x x A k 1 x sin 1 x 0 0 0 1 2 1 x 0 2 1 x x 1 log2 1 log2 0 u 0 1 log2 0 x x 0k 2 1 x 0 0 x 2 0 x 2 2 19 x 2 0 2 0 x 2 x 2 k x 2 0 0 0 p x2 x x 1 2 x x2 x x 1 2 x x2 x 2 x2 x x x2 x x 2 x2 x x x2 x 1 x 2x 1 2x l 1 2x 1 2 A 1 2 u 0 A 1 2 x A k x2 x x 1 2 0 0 2 arctgx 2 2 arctgx tg 2 A tg 2 u 0 A tg 2 0 x A k 2 arctgx 2 0 0 2 arctg 1 1 x 2 2 arctg 1 1 x x 1 1 x 0 1 tg 2 0 x 1 x 1 0 1 x k arctgx 2 0 0 1 2 1 1 e 1 x 0 1 1 e 1 x 0 x 0 1 ln 1 1 0 x 0 x k 1 1 e 1 x 0 0 A1 0 x A1 k f x a 0 x A2 k f x b A k a b a f x f x b 0 A1 0 x A1 k f x b 0 A2 0 x A2 k f x b b 2 l k b f x f x b b 2 A max A1 A2 0 x A k 1 f x 1 b 1 b f x f x b 0 A1 0 x x A1 k f x a 0 x x A2 k g x b 0 A3 0 x x A3 k f x M A max A1 A2 A3 0 x x A k f x g x ab f x g x bf x bf x ab f x g x b b f x a 0 1 0 x 0 x a 1 k f x b b 0 x 0 x a 2 k h x b h x 0 x 0 x a k b f x g x h x b g x b 0 x0 B k f x0 A 0 2 lim x a g x 6 A lim x f x 6 A 0 0 x0 a x0 0 A 0 x1 x2 A k f x1 f x2 0 x1 x2 A k f x1 f x2 0 1 y x cosx x 34 y x1 x2 A x1 x2 cosx1 x1 cosx1 x2 cosx1 x1 cosx2 x2 2 x1 l 2 x1 2 A 2 u A 2 0 x1 x2 Ak cosx1 x1 cosx1 x2 A n A k sinn sin n 2 1 0 22 x sinx 34 11 y e f x 3R lim x f x 0 K x R kf x 0 f x 0 y f x T 0 y b x0 R f x0 a 6 0 5 n N k f x0 nT a 6 0 l lim x f x0 nT a 6 0 r n k lim x f x 6 0 fi lim x f x 0g x R f x 0 11 e 4 3 lim x 2 x2 5x 6 x2 12x 20 lim x 2 x2 5x 6 x2 12x 20 lim x 2 x 2 x 3 x 2 x10 lim x 2 x 3 x 10 1 8 5 lim x 1 1 1 x 3 1 x3 lim x 1 1 1 x 3 1 x3 lim x 1 x2 x 2 1 x3 lim x 1 x 1 x 2 1 x 1 x x2 lim x 1 x 2 1 x x2 1 7 lim x 4 1 2x 3 x 2 lim x 4 1 2x 3 x 2 lim x 4 1 2x 3 1 2x 3 x 2 x 2 x 2 1 2x 3 2 x 4 x 2 x 4 1 2x 3 2 x 2 1 2x 3 9 lim x 0 p a bx c dx ac x ac 6 0 lim x 0 p a bx c dx ac x lim x 0 p a bx c dx ac p a bx c dx ac x p a bx c dx ac lim x 0 a bx c dx ac x p a bx c dx ac lim x 0 ad bc x bdx2 x p a bx c dx ac lim x 0 ad bc bdx p a bx c dx ac ad bc 2 ac 23 10 lim x 1 xm 1 xn 1 m ng lim x 1 xm 1 xn 1 lim x 1 x 1 xm 1 xm2 1 x 1 xn 1 xn2 1 lim x 1 xm 1 xm2 1 xn 1 xn2 1 m n 13 e 4 1 lim x 0 sin x 2 4x lim x 0 sin x 2 4x lim x 0 sin x 2 x 2 1 8 1 8 3 lim x 1 sin x2 1 x 1 lim x 1 sin x2 1 x 1 lim x 1 sin x2 1 x 1 x 1 2 4 lim x a sinx sina x a lim x a sinx sina x a lim x a 2sin x a 2 cos x a 2 x a lim x a sin x a 2 x a 2 cos x a 2 cosa 5 lim x a sinsinx x lim x a sinsinx x lim x a sinsinx x sinx x 1 7 lim x 1 1 x 3 x lim x 1 1 x 3 x lim x 1 1 x 3 x 3 1 1 x 3 3 e 8 lim x 2 1 ctgx tgx lim x 2 1 ctgx tgx lim x 2 1 ctgx 1 ctgx e 10 lim x x3 2 x3 3 x3 lim x x3 2 x3 3 x3 lim x 1 2 x3 x3 1 3 x3 x3 lim x h 1 2 x3 x3 2 i 2 h 1 3 x3 x3 3 i3 e 5 24 14 y e f x 3m m a b NO k K4 lim x a f x lim x b f x 3 y ky 4 lim x b f x 3 3 a b S NO bn b1 b2 bn lim x bn b fi f x 3 m a b NO qk K f b1 f b2 f bn q M 0 n N kf bn M u f bn NO k n f bn lim x f bn B y lim x b f x B 0 n N n N kB f bn 0 b bN x b x bN b bN x bm bl k B f b f bN f x f bm B B f x B f x B B k f x A 0 N N n N kan B l f an A 0 a0 B k f a0 A 0 B1 1 a1 B1 k f a1 A 0 B2 1 a2 Bn k f a2 A 0 Bn n a0 Bn k f an A 0 u 3 an w lim n an k lim x f an 6 A fi g lim x f x A 17 A r ny e f x 3 a b k NO K x0 a b 4 f x0 0 lim x x 0 f x f x0 0 lim x x 0 f x 3 f x0 0 f x0 f x0 0 y ky f x0 0 lim x x 0 f x f x0 3 m a x0 S NO an lim x an x0 dfi f an NO k w f x0 f an n f an lim x f an B N N n N k f x B aN m N n m kaN bn m n1 n2 N n1 n2 N an1 bn an2 fi f x NO k f an1 f bn f an2 f an1 B f bn B f an2 B f bn B max f an1 B f an2 B u lim x f bn B r n kf x0 0 lim x x 0 f x B n N kbn 0 a 0 K ka 0 xnk fi f x 3 a b O kf a f a 0 f a C f xn f f xnk f a 4 l f xn f a 4 fi g u lim n xn a 19 4 K9 n ye 4 1 lim n 1 x 1 x 2 1 x4 1 x2x 1 1 x x 1 y 1 x 1 x 1 x2 1 x4 1 x2 x 1 x2 1 x2 1 x4 1 x2 x 1 x4 1 x4 1 x2 x 1 x2 x 1 x 1 x2 1 x4 1 x2 x 1 x2 n 1 x u lim n 1 x 1 x 2 1 x4 1 x2x lim n 1 x2 n 1 x 1 1 x x 1 2 lim n cos x 2 cos x 4 cos x 2n sinx x y cos x 2 cos x 4 cos x 2n 2nsin x 2n 2n 1cos x 2 cos x 4 cos x 2n 1 sin x 2n 1 2n 2cos x 2 cos x 4 cos x 2n 2 sin x 2n 2 2sin x 2 cos x 2 sinx cos x 2 cos x 4 cos x 2n 2nsin x 2n sinx 2nsin x 2n u lim n cos x 2 cos x 4 cos x 2n 2nsin x 2n 27 lim n sinx 2nsin x 2n lim n sinx x x 2n sin x 2n sin x 3 lim x 0 x h 1 x i 1 y x 6 0k 1 x 1 0 kx 1 h x 1 x i 1 x 0 k1 h x 1 x i Y n klim x 0 x h 1 x i 1 4 lim x a x a h b x i b a y x 6 0k x a h b x i x a b x nb x o b a x a nb x o fi x 6 0 k nb x o 0 b 0 y x 0 x a k0 x a 0 0 x 0 x a A 2 lim x a g x lim x a g x A 0 0 x a x A 3 lim x h x lim x h x A 0 B 0 x x B kh x 0 B 0 x B ks x 0 0 x a x a kf x 0 B 0 x A 2 y 1 lim x 0 3x 1 x2 y 0 x 1 2 x 2 A l 1 2 x 2 A x 0 min