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最新2019-数列极限-23-PPT课件_图文

Ch2
2.1
() ()()

()
Def :1,2,

a,a,a ,a,

1 2 3,

n

{a } n

,a

n



n a n .
2,4,8, ,2n, ; { 2 n }

12,14,18,,21n,;

{

1 2n

}

1,1,1, ,(1)n1, ; {(1)n1}

2,1,4, ,n(1)n1, ;{n (1)n1 }

23

n

n

a nf(n )n ( N )(.)

()

S

S.

A16,

A212,

A324, , An62n-1,

A123

n, An

S.

n(n) ,{An} S(AnS )

Def n N , , {an} a n a, {an}a , a{ an},
l n ia n m a a n a ( n )
{an} , ln i a m n . 1 n
(1){1};(2){(1)n1};(3){2n}; (4)41n;(5){ln1n};(6)2n1

( 1 ) { 1 }( 2 ; ) { 1 ( ) n 1 }( 3 ; ) { 2 n }( ; 4 ) 4 1 n ; ( 5 )l1 { n n }( 6 ) ; 2 n 1

(1)lim 11. n

(2)lim (1)n1 n

.





(3)lim 2n lim 2n..

n

n

(4)ln im41n 0



(5)lim1 0 nlnn

1
(6)lim2n 1 n

:

(1)lim qn0(q1) n
1
(3)lim an 1(a1) n

(2)ln i m n1k 0(kR)
(4)ln imnnn! 0

: n, x n
?,?

:

n, xn1 (1 n ) n11 .

: ""? .



xn1(1)n1

1 n



1 n

1 , 100

1 1 , n 100

n 10 ,0xn

1 1 , 100

1 , 1000

n 10 0,0xn

1 1 , 1000

1 , 10000

n100 0, 0 xn

1 1 , 10000

0, nN([1] ) , xn1.

( ),N,n Nxn , xn a , a xn,xna ,
lnim xn a, xn a (n).
,.
1 . x na x n a ;
2.N .

N : ln i x m na 0 , N 0 , n N , x n a.
:; : .
:
a 2 a x 2 x 1 xN1 a xN2 x 3 x
nN , xn (a , a) ,
( N ) .

2 x n C ( C ) , l n ix n m C .
0,n,
xn C CC 0, , ln im xnC.
:.
: , 0,N,N.

3 li q m n0 ,q 1 . n

0, q0, lim qnlim 00;

n

n

0q1, xn0qn, nlnqln,

n ln , ln q

N [ln], n N , lnq

q n0, lim qn0. n

{a n} .

4

(1)lim2n1; n n3
(3)lim( n2 n1); n
(5)ln im 242nn1 3n31n.

(2)ln im 3nn332nn21; (4)lim[ln2(n1)lnn];
n



(1)lim2n1 n n3

2 lim n 1

1
n 3

2.



(2)ln i m3nn332nn21

n lim

3



2 n2

n 1

1

1 n3

3.

n

lim 2 n 1 n n 2 n

2 1

lim

n 2

n 1 1

n

(3) li(m n2n1) n lim 3 n n2 n1 0.



li(m n 2 n 1 ) 0
n

(4 )li[m l2 n n 1 ) ( ln ] n





ln 2 n (1 )ln ln2n n

1

ln

2



n1

li[m l2 n n 1 ()ln ]limln21ln2

n

n n

(5)ln im 242nn1 3n31n



4n 3n1 22n1 3n



1



3

3 4

n

2





3 4

n



4n 3n1 nl im22n1 3n



lim
n



1 2

3

3 4 3

n n

4



1 2

2.2
:n a n f(n ) . y f(x ) , D , f(x ) : x D f( x) .

f(x ) 2 x 2 D ( , )
y f (x)

x0

x

(1) x , x ;f(x)

(2) x , x ;f(x)

(3)x , x ;

f(x)

(4 )x x 0 x x 0, x x0 ; f(x) f(x0) (5 )x x 0 x x 0, x x0 ; f(x) f(x0) ( 6 ) x x 0 x x 0 , x x0; f(x) f(x0)

x X6

x X , f(x)

A, x X , f(x)A ,

A f(x) x X ,

li f ( x ) m A f ( x ) A ( x X ) x X
f (x ) x X l if m (x ) . x X



lif m (x ),lif m ( x ),lif( m x ),

x x 0

x x 0

x x 0

lifm (x ),lif m (x ),lif( m x ).

x x

x


1 (), X,x X x,f (x)f (x) A , Af (x)x , limf (x) A f (x) A(x )
x
"X"lim f(x)A x
0 , X 0 , x X , f ( x ) A .

:
10.x : lim f(x)A x
0 , X 0 , x X , f ( x ) A .
20.x : lim f(x)A x
0 , X 0 , x X , f ( x ) A .

:l im f(x )A lif( m x ) A lif( m x ) A .

x

x

x

: l im f(x)c, yc yf (x) x

.

:

X

A
X

xX xX ,y f(x) y A, 2 .



: yf(x) x x0 , f(x) A .

f(x )A f( x )A ;

0 x x 0 x x 0 .





x0

x0

x0 x

x0 , x x0 .

2 (

),,

0 x x0 x,f (x) f (x) A ,A

f (x)x x0,

lim f (x) A
xx0

f (x) A(x x0)

"" 0,0,0 xx0 ,

f (x)A.

1. f(x) x 0 ;

2. .

:



x



x


0



, y f ( x )



y A ,

2

y
A
A
A
.o

yf(x)



x0 x 0 x0

x

, , .

(1)x l ix0m f(x) x x0. x x 0 f(x ) f(x ) x 0 .

f(x ) 2 x 2x 0 1 x 0

lim f(x)2
x 0

y f (x)
x

(2 ) x l x i0 m f(x ) x x 0 (f( x 0 0 ) ); x l x i0 f m (x ) x x 0 (f(x 0 0 )) .

(3)l im f(x ) . x
(4)limf(x) xX
l if ( m x ) , x " X " f ( x ) " " x X
f (x) x X , .

0,0,x0 xx0 , f (x)A.
x lx 0 i 0m f(x ) A f(x 0 0 ) A . (x x 0 )
0,0,x0 xx0 , f (x)A.
x lx 0 i 0m f(x ) A f(x 0 0 ) A . (x x 0 )
: x l x 0 i f ( x ) m A f ( x 0 0 ) f ( x 0 0 ) A .

limx . x0 x

y

limxlimx x0 x x0 x lim (1)1
x 0

1

o

x

1

x lim

limx

lim11

x x x0

x0

x0

, limf(x). x0

1. :( )

1 11 1 11 ( 1 )li, m li, m li,m li, m li,m li.m
x x x x x x x x x x 0 x 0 x 0

y

lim1 0, lim1 0, lim1 0,

y 1 x

x x

x x

x x

1 lim , x x0

1 lim x x0



,

1 lim x0 x



.

O

x

(2 )lilm x n , lilm x n , lilm x n

y

ylnx

x x 0

x 1

O
lim lnx , lim lnx . lim lnx0. 1

x

x

x0

x1

y

(3 )lie x m ,lie m x , lie m x , lie x m x 0 x x x

limex 1 limex 0, limex ,

x0

x

x

y ex

x

O

limex .
x y

(4)lis m ixn ,lis m ixn

x 0

x

lis m ixn si0n 0
x 0

2

O 2

ysinx



x

x x ,sinx 1

1," " .

l is m ix ,nlis m ix ,nlis m ixn .

x

x

x



1 ln im an

2

lif m (x ),lif m ( x ),lif( m x ),

x x 0

x x 0

x x 0

lifm (x ),lif m (x ),lif( m x ).

x x

x

limf(x) !
xX

(1) f(x), x0 f(x)
, xx0 f(x)f(x0),
x l ix0 m f(x)f(x0)

(2) (Th2.1)

limf(x)A

xx0



lim f(x)lim f(x)A

x x 0

x x 0

limf(x)A
x

lim f(x )lim f(x )A

x

x

1. :( )

( 5 )lia m rx ,clt ia a m r n x ,c lt ia m arx n .cta

x

x

x

lim arctxa n,

x

2

y
2
yarctxan

lim arctxan,

x

2

O

x

2

l im arcxt an . x

2 . f(x ) x 1 ,,x x 0 0 , lx 1 ifm (x ) l x 0 ifm (x ).

x 1 ,f(x ) x ,

limf(x) limx1.

x1

x1

x=0

limf(x) limx 0,

1

x0

x0

limf(x) lim1 1,

x0

x0

1

li f m (x ) lif m (x ), lim f(x) .

x 0

x 0

x 0

: x 0 ,(1 ) f(x ) 1 1

x 0 x 0

(2)f(x)x .

2.2 limf(x)A, A0( A0), x x0
x0 x f(x)0( f(x)0). (1)
2 .3 li f m (x ) A , f(x ) 0 ( f(x ) 0 ), x x 0
A 0 ( A 0 )(.2)

2.3
a n f(n ) , f(x ) y n ,x X
y A, y A, lim yA( ).

2.7 y M , y y M, y .
: xn, M , n, xn M , xn, , .
xn [M,M] .

2.4 y

y .( )

:f (x ) 1 x

x 0 x 0 l ifm (x ) .

x 0

x 0

. . . 2.5 .


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