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1、 10 1 1 1 P n 1 cosn n n 1 n p P k n 1 cosk k k 1 n p P k n 1 1 k k 1 n p P k n 1 1 k 1 k 1 1 n 1 1 n p 1 0 N 1 n N n p P k n 1 cosk k k 1 1 n 0 P n 1 an P n 1 bn N N n N n p P k n 1 ak n p P k n 1 bk n p P k n 1 ak n p P k n 1 uk n p P k n 1 bk n p P k n 1 uk 0 lim n S2n lim n S2n 1 A N 0 n N S2n A
2、 S2n 1 A N Sn A 0 P n 1 un N n N n p P k n 1 un N 2n u2n 2 2n P k n 1 uk 2n 1 u2n 1 2n P k n 1 uk 2n 1 P k n 1 uk 1 4 P n 1 1 n1 1 n lim n 1 n1 1 n 1 n 1 P n 1 1 n 5 P n 1 n2 3 1 n n lim n n q n2 3 1 n n 1 3 6 P n 2 n lnn n lim n n q n lnn n 0 7 P n 1 1000n n lim n 1000n 1 n 1 1000 n n 0 8 P n 1 n 2
3、 2n lim n n 1 2 2n 2 n 2 2n 1 4 9 P n 1 1 3n n 1 n n 2 lim n n q 1 3n n 1 n n 2 e 3 0 p 1 R 2 1 x lnx p lnx 1 p 1 p 2 p 1 R 2 1 x lnx p lnlnx 2 0 p 0 lim n 1 n lnlnn q 1 n lnn 1 2 lim n lnn 1 2 lnlnn q lim x x1 2 lnx q P n 3 1 n lnn 1 2 3 P n 1 un P n 1 u2 n P n 1 un lim n un 0 un un M u2 n Mun P n
4、1 u2 n P n 1 1 n 2 P n 1 1 n 4 P n 1 a2 n P n 1 b2 n P n 1 anbn P n 1 a n bn 2 P n 1 an n P n 1 a 2 n b2n anbn 1 2 a 2 n b2n an bn 2 3 2 a2 n b2n P n 1 anbn P n 1 a n bn 2 bn 1 n P n 1 an n 5 P n 1 un P n 1 vn 1 P n 1 u n vn 2 P n 1 u n vn 3 P n 1 unvn 1 un vn un P n 1 u n vn 2 un vn 1 n P n 1 u n v
5、n 0 3 un vn 1 n P n 1 unvn 6 lim n nun l 0 l 0 n un 0 un 0 lim n u2 n 1 n2 l2 P n 1 1 n2 P n 1 u2 n lim n un 1 n l P n 1 1 n P n 1 un 10 3 1 1 P n 1 1 n 1 2n 2 P n 1 1 n 1 2n 2 P n 1 1 2n 2 2 P n 1 1 n 1 2n 1 p p 0 1 2n 1 p 0 P n 1 1 2n 1 p p 1 p 1 3 P n 2 1 n nlnn 1 n lnn 0 P n 2 1 nlnn 4 P n 1 1 n
6、 n 1 n P n 1 1 n 1 n P n 1 1 n 1 n lim n n 1 n 1 n 1 P n 1 n 1 n 6 P n 1 1 n n 3n2 lim n n 1 3 n 1 2 n 3n2 lim n n 1 32n 1 0 4 7 P n 2 1 n 1 n sin n lim n 1 n sin n 1 n2 8 P n 1 1 n 1tan n 2 0 P n 1 n n 1un 1 np n n 1 P n 1 un P n 1 un np P n 1 n n 1un 3 P n 1 cosn np 0 1 0 1 P n 1 1 np cosn np 1 np
7、 P n 1 cosn np ii 0 x0 x x0 nx0 x P n 1 an nx P n 1 an nx0n x0 x 6 P n 1 un P n 1 2n 1 n un 2n 1 n un 2 un P n 1 un P n 1 2n 1 n un 10 4 5 1 1 P n 1 lnx n lnx 1 3 1 xn sin 3n 1 x n 3n 1 3 1 3 2 1 fn x 1 2n x2 x lim n fn x 0 fn x 0 1 2n lim n 1 2n 0 2 fn x x4 e n x lim n fn x x4 x2 fn x x2 e n x4 e n
8、 x2 e n e n e n 2 lim n e n 2 0 3 fn x ln 1 x2 n2 a l x l b x lim n fn x 0 a fn x 0 x2 n2 l2 n2 lim n l2 n2 0 l l b xn n lim n f xn 0 ln2 4 fn x n2x 1 n2x 0 x 1 lim n fn x 1 xn 1 n2 lim n f xn 1 1 2 3 1 P n 1 1 n n x2 n2 x 1 n n x2 n2 n n2 1 n3 2 M 2 P n 1 xn n xn 1 n 1 1 x 1 P k n 1 x k k xk 1 k 1
9、x n 1 n 1 1 n lim n 1 n 0 3 P n 1 sin nx 1 x2 n2 3 x sinnx 1 x2 n2 3 1 n3 M 4 P n 1 x 1 4n4x2 x 6 1 4n4x2 4n2 x x 1 4n4x2 1 4n2 M 5 P n 1 x2 1 x n x P k n 1 x2 1 x k x 1 x n x 1 nx 1 n lim n 1 n 0 6 P n 1 sinx sinnx n2 x2 0 x 2 1 n2 x2 1 n lim n 1 n 0 1 n2 x2 0 1 n2 x2 n n P k 1 sinx sinkx cos x 2 c
10、os x 2 cos n 1 2 x 2 7 P n 1 1 n 1x2e nx2 x 0 g x i xn 3n 1 P k n 1 2ksin xn 3k 2 n 1sin1 sin1 ii x M M 2nsin x 3n 2 n x 3n M 2 3 n M P n 1 2nsin x 3n M M iii 2nsin x 3n 2 3 ncosx 3n 2 3 n M P n 1 2 nsinx 3n g x 6 x P n 1 1 nx 1 0 P n 1 lnn nx 1 0 x 1 7 i x 1 1 nx 1 n1 M P n 1 1 nx 1 ii lim n lnn n
11、2 0 M 0 lnn n 2 0 lim n n q 1 nan 1 a a a x a x a a a 3 P n 0 1 nx2n 1 2n 1 2n 1 lim n x2n 3 2n 3 2n 3 x2n 1 2n 1 2n 1 0 8 5 P n 0 x2n 1 x2 0 n N ln 1 an 1 lnn an 0 P n 1 an n N ln 1 an 1 lnn 1 an n1 an 1 n1 P n 1 an 9 y f x x0 a x0 a a 0 f n x M M f x P n 0 1 n f n x0 x x0 n x0 a x x0 a f x x x0 Rn x 1 n 1 f n 1 x0 x x0 x x0 n 1 x x0 a x0 a Rn x Man n 1 lim n Rn x 0 f x f x 11
