大一上数分课堂课件第二章

§2.5函数的极限(2)四、极限存在定理1.

夹逼定理d

0(1)

g(

x)

£

f

(

x)

£

h(

x),如果当

x

˛

U

o

(

x

)

(或

|

x

|>

M

)

时,(2) lim

g(

x)

=

A,xfi

x0(

xfi

¥

)lim

h(

x)

=

A,xfi

x0(

xfi

¥

)那么

lim

f

(

x)

存在,

且等于A.xfi

x0(

xfi

¥

)证明：$d1

>0,当0

<|

x

-x0

|<d1时,A

-e

<g(x

)<A

+e;$d2

>0,当0

<|

x

-x0

|<d2

时,A

-e

<h(x

)<A

+e;取d

=min(d1

,d2

),当0

<|

x

-x0

|<d时A

-

e

<

g(

x)

£

f

(

x)

£

h(

x)

<

A

+

e;故

|

f

(

x)

-

A

|<

e."

e

>

0,–

¥

)中有数列xn

(„

a),

使得n

fi

¥

时xn

fi

a.则称数列{f

(xn

)}为函数f

(x)当x

fi

a时的子列.设在过程xfi

a(a可以是x

,

x

+

,

x

-

,

¥

或0

0

0定义nfi

¥x

fi

a子列{f

(xn

)},都有lim

f

(xn

)=A.lim

f

(x)=A x

fi

a时,f

(x)的任何Heine定理(函数极限与数列极限的关系)2.

Heine定理证\对上述d

>0,

$N

>0,使当n

>N时,恒有0

<

xn

-

x0

<

d.从而有f

(xn

)-A

<e,x

fi

¥故

lim

f

(

xn

)

=

A.nfi

¥

lim

f

(

x)

=

Ax

fi

x0\

"

e

>

0,

$d

>

0,

使当0

<

x

-

x0

<

d时,

恒有f

(

x)

-

A

<

e.又

lim

xn

=

x0

且

xn

„

x0

,xfi

x0>

0,假设

lim

f

(

x)

=

A不成立，则必$e0*对"

n

˛

N

,的1n0存在满足0

<|

x

-

x

|<n点x

，使得|

f

(

xn

)

-

A

|‡

e0

>

0.即找到了一个数列{

xn

|

xn

„

x0

},

xn

fi

x0

,但是lim

f

(xn

)„A.nfi

¥函数极限存在的充要条件是它的任何子列的极限都存在,且相等.例7证明lim

sin

1

不存在.xx

fi

0npn证

取

{x

}=

1

,nfi

¥lim

xn

=

0,21,

且

xn

„

0;p4n

+

1n取{x¢}=nfi

¥

lim

xn

=

0,且xn

„0;xnnfi

¥nfi

¥而lim

sin

1

=

lim

sin

np2=

lim

sin

4n

+

1pnfi

¥x¢nnfi

¥而lim

sin

1=

lim1

=

1,nfi

¥二者不相等,故lim

sin

1

不存在.xxfi

0=

0,xy

=

sin

1lim

f

(x)存在xfi

x00

<|

x1

-x0

|<d,0

<|

x2

-x0

|<d,都有f

(x1

)-f

(x2

)<e.证明：lim

f

(

x)

=

A,x

fi

x0当0

<

x

-

x0

<

d时,|

f

(

x)

-

A

|<

e

2

.特别：取0

<|

xi

-

x0

|<

d

,

i

=

1,2

则2

21

2

1

2f

(

x

)

-

f

(

x

)

£

f

(

x

)

-

A

+

f

(

x

)

-

A

<

e

+

e

=

e."

e

>

0,

$d

>

0,3.

柯西准则(Cauchy)"e

>0,

$d

>0,

使得"x1

,x2

,如满足①任取一个{

xn},

lim

xn=

x

0

(

xn

„

x

0

)n

fi

¥对

d

>

0,

$N

˛

N

*

,当m

,

n

>

N时，0

<

xn

-

x

0

<

d

,0

<

|

xm

-

x0

|

<

d\

|

f

(

xn

)

-

f

(

xm

)

|<

e\{f

(xn

)}是Cauchy列.\lim

f

(xn

)=l

x

存在.nfi

¥(yn

„x

0

),则n

fi

¥{f

(yn

)}也是Cauchy列.\lim

f

(yn

)=l

y

存在.nfi

¥② 另一个

{

yn

},

lim

yn

=

x

0nfi

¥lim

zn

=

x0

且(

zn

„

x0

),\

lim

f

(zn

)

=

l.nfi

¥③往证：

l

x

=

l

y将x1

,y1

,x2

,y2

,,xn

,yn

,组成{zn

}.nfi

¥据Heine定理,知lim

f

(

x)

=

l.xfi

x0由于{xn

},{yn

}是{zn

}的子列,所以{f

(xn

)},{f

(yn

)}是{f

(zn

)}的子列,\

l

x

=

l

y

=

l.\

对任意收敛于

x0的数列{

xn

},xn

„

x0

,都有lim

f

(xn

)=l.五、函数极限的性质1.

唯一性若

lim

f

(

x)存在，则极限必唯一.x

fi

x0证明：任取xn

fi

x0

(

xn

„

x0

)，由Heine定理lim

f

(xn

)=lim

f

(x

),由数列极限唯一性可知.nfi

¥

x

fi

x0证明：取e

=

1,

$d

>

0,0

<|

x

-

x0

|<

d时|

f

(

x)

-

A

|<

e

=

1.|

f

(

x)

|=|

f

(

x)

-

A

+

A

|£|

f

(

x)

-

A

|

+

|

A

|£

1+

|

A

|=

M

.0lim

f

(

x)存在,

则f

(

x)在x

的邻域U

o

(

x

)内有界.d

0xfi

x02.

有界性定理(保序性)设lim

f

(

x)

=

A,

lim

g(

x)

=

B.xfi

x0

xfi

x0若$d

>

0,"

x

˛

U

o

(

x

,d

),

有f

(

x)

£

g(

x),则A

£

B.03.

不等式性质0xfi

x0f

(x)‡0(或f

(x)£

0),则A

‡0(或A

£

0).若lim

f

(

x)

=

A,且$d

>

0,当x

˛

U

0

(

x

,d)时,推论xfi

x0若lim

f

(

x)

=

A,

且A

>

0(或A

<

0),定理(保号性)则$d

>

0,当x

˛

U

o

(

x

,d

)时,

f

(

x)

>

0(或f

(

x)

<

0).0证明：20

0Ao<|

x

-

x

|取e

=

,

$d

>

0,0

<

d

(即x

˛

U

(

x

,d

)),20

<

A

<

f

(

x)

<

3

A.2

2|

f

(

x)

-

A

|<

A

,推论

设lim

f

(

x)

=

A,

lim

g(

x)

=

B,

且A

<

Bxfi

x0

xfi

x0则$d

>

0,"

x

˛

U

o

(

x

,d

),

有f

(

x)

<

g(

x).0六、极限运算法则⒈四则运算法则定理(3)lim

f

(

x)

=

A

,g(

x)

Blim[

f

(

x)

–

g(

x)]

=

A

–

B;lim[

f

(

x)

g(

x)]

=

A

B;其中B

„0.设lim

f

(x)=A,lim

g(x)=B,则⒉复合运算

f

g(t

)

=

f

[

g(t

)]定理设

lim

f

(

x)

=

A, lim

g(t

)

=

x0

,xfi

x0

t

fi

t0且在U

o

(t

)内g(t

)

„

x

,d

0

0则lim

f

g(t

)]=

lim

f

(

x)

=

A.t

fi

t0

x

fi

x0解释：相当于计算中进行变量替换，g(t

)=x证明：

lim

f

(

x)

=

A,xfi

x0\

"

e

>

0,

$s

>

0,当x

˛

U

o

(

x

,s

)时,0|

f

(

x)

-

A

|<

e.对上述s

>

0,

$d

>

0,

当t

˛

U

o

(t

,d

)时,0|

g(t

)

-

x0

|<

s

,\

g(t

)

˛

U

o

(

x

,s

),

\

|

f

[g(t)]-

A|<e.0u

=

01,

u

„

0,y

=

f

(u)

=

0,xu

=

g(

x)

=

x

sin

1lim

g(

x)

=

0,lim

f

(u)

=

1,xfi

0

ufi

0lim

f

(g(x))不存在xfi

0且在U

o

(t

)内g(t

)

„

x

,d

0

0⑴例8、lim3-

1x

2

-

1xfi

1

x2=

lim(

x

-

1)(

x

+

1)xfi

1

(

x

-

1)(

x

+

x

+

1)23=⑶⑵

lim2+

x

+

4x

2

+

x

+

9xfi

¥

x2

9+

1

+

4x x

21

+

x

+

x

2=

limxfi

¥1=

1lim

x

-

3

=

0xfi

3

x

+

2⑷lim45

2==

lim+

7

x

+

2x2

+

2

x

+

3

32+

7

x

+

2

xx3

+

2

x2

+

3

xxfi

0

xxfi

0

x⑸limxfi

0limun

-

1x

ufi

1

u

-

1=

lim(un-1

+

un-2

+

+

u

+

1)

=

nufi

1(

x

+

1)n

-

1

1

+

x

=

uAC七、两个重要极限(1)xlim

sin

x

=

1x

fi

02设单位圆

O,

圆心角—

AOB

=

x, (0

<

x

<

p

)于是有sin

x

=

BD,

x

=

弧

AB,

tan

x

=

AC

,xoBD作单位圆的切线，得DACO

.扇形OAB的圆心角为x

,DOAB的高为BD

,\

sin

x

<

x

<

tan

x,x即cos

x

<sin

x

<1,p2当0

<x

<p

时,2

x22x<

2(

2

)x

2=

2

,2=

0,x

2

limx

fi

0\

lim(1

-

cos

x)

=

0,x

fi

0\

limcos

x

=

1,x

fi

0x又

lim

1

=

1,

\

lim

sin

x

=

1.x

fi

0

x

fi

0上式对于-2

<x

<0也成立.0

<

1

-

cos

x

=

2sin例9⑴xxfi

0⑵xlim

sin

x

=1的应用x

fi

0x

cos

x1lim

tan

x

=

lim

sin

xxfi

0=

1xfi

0

sin

bx

ax

sin

bx

b(lim

cos

x

=1,由刚才过程可知)x

fi

0lim

sin

ax

=

lim

sin

ax

bx

axfi

0=

ab(2)xx

fi

¥lim(1

+

1

)

x

=

e)

}nn1n曾考虑数列{

x

}

=

{(1

+lim

xn

存在.nfi

¥nnfi

¥记为lim(1

+

1

)n

=

e(e

=

2.71828)当x

‡1

时,有[x]£

x

£

[x]+1,x

fi

+¥

x

fi

+¥x

fi

+¥而

lim

(1

+

1

)[

x

]+1

=

lim

(1

+

1

)[

x

]

lim

(1

+

1

)

=

e,1

1[

x]

[

x]

[

x]1[

x]

+

1

[

x]

+

1[

x]

+

1lim

(1

+x

fi

+¥)[

x

]+1=

lim

(1

+x

fi

+¥)[

x

]lim

(1

+x

fi

+¥)-1

=

e,x\

lim

(1

+

1

)

x

=

e.x

fi

+¥1[

x]

+

1(1

+)[

x

]

£

(1

+

1

)

x