高数极限运算法则

1、2021/3/101 2021/3/102 ,)(lim,)(limBxgAxf )()(limxgxf)(lim)(limxgxfBA )()(limxgxf)(lim)(limxgxfBA )( )( lim xg xf )(lim )(lim xg xf B A )(lim)(limxfCxfC nn xfxf )(lim)(lim 2021/3/103 .lim 1 1 10 0 nn nn xx axaxaxa , 0 0 limxx xx ,lim 0 0 nn xx xx .limlimlim 1 1 10 000 nn xx n xx n xx axaxaxa . 01 1 0

2、100nn nn axaxaxa 2021/3/104 1 2 2 3 x x x lim 4 11 13 232 3 1 lim 3 x x x 9 34 lim 2 2 3 x xx x)3)(3( ) 1)(3( lim 3 xx xx x 3 1 2021/3/105 . 45 32 lim 2 1 xx x x 32 45 lim 2 1 x xx x 0 312 41512 45 32 lim 2 1 xx x x 2021/3/106 . 125 934 lim 2 2 xx xx x ,分子时x.分母 2 2 11 11 25 934 lim x x x x x , 2 x 5

3、 4 2021/3/107 . 125 94 lim 2 xx x I x 2 2 12 5 94 lim xx xx I x 0 x,. , 2 x 2021/3/108 nmba,0( 00 mn 当 m mm x axaxa 1 10 lim n nn bxbxb 1 10 , 0 0 b a ,0 , mn 当 mn 当 2021/3/109 . 23 123 lim 23 2 xxx xx x 5 32 861 lim. 432 x xx xxx -+ + 2021/3/1010 . )75( )53() 12( lim 50 3020 x xx x . 5 32 50 3020 0

4、 0 b a 2021/3/1011 )1(lim 2 xxx x x)( xx1 2 xx x x 1 lim 2 1 2 = 1 1 1 1 lim 2 x x 2021/3/1012 ). 21 (lim 222 n n nn n n ) 21 (lim 222 n n nn n 2 21 lim n n n 2 ) 1( 2 1 lim n nn n ) 1 1 ( 2 1 lim n n . 2 1 2021/3/1013 )(lim 0 xf xx Auf u )(lim .lim )(lim 00 Aufxf uuxx , 0 xx ,xuufy ,lim 0 0 ux xx ,

5、lim 0 Auf uu 0 uu 2021/3/1014 . 9 3 lim 2 3 x x x 9 3 )( 2 x x xfy 9 3 2 x x uuy与 . 6 1 lim 9 3 lim 6 1 3 3 u x x u x . 6 1 lim 9 3 lim 3 2 3 u x x xx 2021/3/1015 ( )yf u 0 u,)( 0 Auf 6 6 6 1 3 1 lim 9 3 lim 9 3 lim 3 2 3 2 3 xx x x x xxx )(lim()( )(lim 00 0 xfufAxf xxxx 2021/3/1016 0 1 lim, x x 2 1

6、 1 1 lim x x x e 2 1 1 lim 1x x x e 2 e 1 0 lim. x x e 1 u x lim0 u u e 2021/3/1017 . 1 1 lim 1 x x x ,xu , 1lim 1 u x 1 1 1 1 2 u u x x 1 u ) 1(lim 1 u u 2 1 1 lim 1 x x x 1 ) 1)(1( lim 1 x xx x ) 1(lim 1 x x 2 2021/3/1018 32 1 4 lim 1 x xaxx x 1 lim(1)0, x x 32 1 4 lim 1 x xaxx l x 极极限限 32 1 lim(4)0, x xaxx 32 1 4 lim 1 x xaxx x 1 (4)(1)(1) lim 1 x xxx x 1 lim(4)(1)10, x xx 2021/3/1019 2021/3/1020 )(xf bxaxxxf 23 22)( x xf x )( lim3 0 0,3ba ,2 2)( lim 2 3 x xxf x ,3 )( lim 0 x xf x . )(xf )