# D1 6极限存在准则

(1)

## 两个重要极限

(2)

(3)

a z

y n

n n

n  

lim

lim )

2 (

1. 夹逼准则 ( 准则 1)

) ,

2 , 1 (

) 1

( ynxnzn n  

a xn

n

lim

 0, N1,

nN1 时 , yn  a

nN2 时 , zn  a

N max

N1 , N2

, 则当 nN 时 , 有

,

  

y a

a n a

zna

, 由条件 (1) a

ynxnzn  a

xn  a

, lim xn a .

n

2, N

(4)

1

lim 21 2 2  

 

 

 

 

n n

n

n n

n



 

 

 

n

n n

n n2 2 2 1

2 1

1 

n

n

n

2 2

22 n

n

n n

n

n 2

2

lim

nn

1

lim 1 1

2

2

lim n n

n 1 2

lim 1

nn

1

n n

 lim 

 

 

 

n

n n

### 

n2 2 2

1 2

1

1  1

(5)

2. 单调有界数列必有极限 ( 准则 2 ) M x

x x

x12   nn1   

m x

x x

x12    nn1   

) (

lim xn a M

n  

) (

lim xn b m

n  

xn xn1 M

x1 x2 x

m xn1xn x2 x1 x a

b

(6)

xn

n n n

x  (1 1)

1n 1!n 12

! 2

) 1 (

n n

n

( 13)(! 2) 13

n n

n

n

nn

n

n n n

n 1

!

) 1 (

) 1

(

1 1

1

1!

n

n

n2

nn1

1

! 21

n

31!

1n

2n

(7)

1 1 xn

1

1!

n

n

n2

nn1

1

! 21

n

31!

1n

2n

1 1 1

xn 21!

n11

31!

n11

n21

11 21 1

! ) 1 ( 1

n n n

nn

, 2 , 1

1 (  

x n xn n

 (1 1 nn) 1 1

xn 21!

! 31

n1!

(8)

###  

xn

, n n e

n  

(1 ) lim 1

e 为无理数 , 其值为

59045 7182818284

.

 2 e

 (1 1 nn) 1 1

xn 21!

! 31

n1!

1 1 21 2

2

1 1

2 1

###  

n

 3

21 2

1

1 1 1

 

n 1

2 3 1

n

(9)

2. 函数极限存在的夹逼准则 定理 2.x  (x0 ,

) 时,

A x

h x

g x x

x

x  

( ) lim ( ) lim

0 0

, ) ( )

(x h x

gf (x) 

A x

x f

x

( ) lim

0

) 0 ( x  X

)

(x (x )

) (x

(10)

sin 1

cos  

x x x

### 二、 两个重要极限

sin 1 lim

.

1 0

x

x

x

12sin x12 x21 tan x

, 0 ( 2

x 时，

) 0

(  x2 ,

1 cos

lim0

x

x sin 1

lim0

x

x

x

△AOB 的面积＜ ＜△ AOD 的面积

D C

B x A

o

x x

x

cos 1 1 sin 

(11)

x

x

x

x

(12)

n n

n R

sin cos lim 2

n

n A

lim

n

n n n R n

A2 sin cos

R2

) (

) ( lim sin

0 )

(

x

x

x

(13)

2 x x e

x  

(1 ) lim 1

z  

)1

1 ( lim0

(14)

x

x

xlim (1 1x) t t

t

(1 ) lim 1

lim 1

t t

t) 1

(  1 e

 1

)

( x x e

x  

(1 1 )

1 1

 

x x e

x

(15)

] [

lim

x

x

x

)2

sin 1

(

lim 2x x

x

) sin 1

(  2x

e

x 2 x

sin2 2x

sin 1

(16)

(1) 数列极限存在的夹逼准 则

(17)

2. 两个重要极限 sin 1

lim )

1

( 0

e

1 )

1 ( lim )

2 (

1

) 1

( lim0

(18)

; _____

lim sin .

1 

x

x

x 1 ____ ;

sin lim

.

2 

x x

x

; 1 ____

sin lim

.

3 0

x x

x 1) ____;

1 ( lim .

4  

n

n n

1

Updating...

## References

Related subjects :