工科數(shù)分課件第二章2節(jié)簡

性質(zhì)4.1.（唯一性limfx)存在，則極限必唯一xfi性質(zhì)4.2.（局部有界性dlimfx)存在則fx)在x0的鄰域Uox內(nèi)有界d0xfi性質(zhì)4.3limfx)Alimgx)B,xfi xfi若A>

0f(x)>g(

時,fxgx),0A?性質(zhì)4.4limfx)A,且A0(或Axfi性質(zhì)

時,fx0(或fx0 x?Uox時 g(x)￡f(x)￡h(limg(x)=xfi

limh(x)=xfilimfx存在且等于xfi性質(zhì) x?Uox時 g(x)￡f(x)￡h(limg(x)=xfi

limh(x)=xfilimfx存在且等于xfi例3

limsinx xfi sinx<x<tano 即cosx<sinx< o x上式對于

2

xp時2220<1-cosx=2sin22x

< <2

=x xfi

= \lim(1-cosx)=xfi\limcosx=

又lim1= \limsinx=xfi

xfi xfi 例1求極限0

=xfi

sin22 =

xfi

sinx2x2

xfi=

limsin2x=limsin2 =xfi

xfi 2limsinkx其中k0xfi limsinx-1)=（因xfi0時,x-1fi0,可 xfi

x-sinx2

xfi

x-

=xfi1

x2-

x+第二個重要極限：m11xexfi 1

1-lim1+ =e=lim1-nfi￥ n nfi￥ n 1[x

1x

1[xxfi+￥:1

￡1

￡1 [x]+1

x

[x] 1x

1-(-x

1-xfi-￥:1+ =1 =1

,tfi x (-x)

t lim1

1 =1j(x)fi￥ j(x) m1+ttfi

=

1(x))(x =j(x)fi1j(x1j(x)filim1+((x)=1e￥1sin =exfi

x2

=lim1

2

=

1

2

=xfi￥ x

xfi￥ x

xfi￥ x

3 1￥3)lim(xfi

x-2)xx+

=lim(1xfi

x+

)x=limxfi￥

x+

3

x+1注注=lim(1xfi

x+

lim-3xfi

=e-定理4.1（函數(shù)極限四則運算性質(zhì)設(shè)limfx)Alimgx)B,lim[f(x)–g(x)]=A–lim[f(x) g(x)]=Alimfx)A 其中Bg( 定理4.2（復(fù)合函數(shù)的極限設(shè)limf(x)= limg(t)=x0xfi tfid且在Uo(t內(nèi)g(txd則limfgtlimfx)tfi xfi 定理4.3(海涅定理)設(shè)函數(shù)f(x)定義在U0( 則limfx)Axfi "xn U0xfix limf(xn)=nfi注

limf(x)=xfi>0,$d0,使當(dāng)0

x-

<時,f(x)-A<又limxnx0xnx0nfi對上述d0,$N0,使當(dāng)nN時0<xn-x0<從而有f(xn)-A< 故limf(xn)=xfilimfx)Axfi *,存在滿足0<|x- x|?0xn|xnx0xnfix0limf(xn)?nfiy=sinx例4證明limy=sinxxfi 取{x}=1nlimxn=nfi

xn 取

4n+ p

x￠=nfi

n 而limsin1limsinnpnfi nfi而limsin1=limsin4n+1p=lim1= nfi

nfi

定理4.4 收斂定理)函數(shù)f(x)定義在U0( 則mfx)A的充分必要條件是：0,$ xfi對x1x20<|x1x0|<0<|x2x0<,fx1fx2 limf(x)=xfi

x-

<時 f(x)-A|<e2特別：取0<|xi-x0|<,i= f(

)-f(x2)

f(

)-A

f(

)-

e+e=.< <任取{xn},滿足 xn=x0 (xn?x0nfi對d>0,$N?N* 當(dāng)m,n>N時0<xn-

<

0<|xm-x0|fxnfxm|<efxn)}是Cauchy列l(wèi)imfxnlx存在.nfi￥同理存在limfynlynfi將x1y1x2y2xnyn,組成{znlimzn=x0