湖南师范大学《高等数学》课件-第3章函数极限

§1

函数极限概念

§2

函数极限的性质

§3函数极限存在的条件

§4两个重要极限

§5无穷小量与无穷大量第三章函数极限湖南师范大学《高等数学》E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410第三章函数极限§1

函数极限概念E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410播放一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、自变量趋向无穷大时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410通过上面演示实验的观察:问题:如何用数学语言刻划函数“无限接近”.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604101、定义：E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604102、另两种情形:E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604103、几何解释:E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例1证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410二、自变量趋向有限值时函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604101、定义：E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604102、几何解释:注意：E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例2证例3证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例3证函数在点x=1处没有定义.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例4证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604103.单侧极限:例如,E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410左极限右极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410左右极限存在但不相等,例5证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410四、小结函数极限的统一定义(见下表)E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410过程时刻从此时刻以后过程时刻从此时刻以后E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410思考题E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410思考题解答左极限存在,右极限存在,不存在.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、填空题:练习题E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410(1),自变量趋于有限值时函数的极限;

作业

3.小结(2),自变量趋于无穷大时函数的极限;(3),函数极限的几何意义;(4),单侧极限的概念;(5),应用函数极限的定义验证函数极限的方法;P47:1,3,4,5,6,7.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

如果f(x)

A(x

x0)

那么f(x)在x0的某一去心邻域内有界

证明有使得则取设);(,0,1,)(lim00ddexUxAxfxxoÎ">$==®.1)(1)(+<Þ<-AxfAxf.);()(0内有界在即dxUxfo

函数极限的性质1.局部有界性E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

如果当x

x0时f(x)的极限存,那么这极限是唯一的

证明，xxfBA时的极限当都是设0,®,)(0,0,0101edde<-<-<>$>"Axfxx时有当则,)(0,0202edd<-<-<>$Bxfxx时有当故有同时成立时则当取，xx)2(),1(0),,min(021dddd<-<=.2)()())(())((e<-+-£---=-BxfAxfBxfAxfBA..即其极限唯一的任意性得由BA=e2.唯一性E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

如果f(x)

A(x

x0)

而且A

0(或A

0)

那么对任何正数r<A(或r<-A),在x0的某一去心邻域内

有f(x)r>0(或f(x)-r<0)

证明);(,0,),1,0(,00ddexUxrArAÎ">$-=Î">使得则取设.)(rAxf=->e有.0的情形类似可证对于<r推论

如果在x0的某一去心邻域内f(x)

0(或f(x)

0)

而且

f(x)

A(x

x0)

那么A

0(或A

0)

3.局部保号性E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410证明).(lim)(lim),()();()(),(00'00xgxfxgxfxUxgxfxxxxxx®®££®则内有极限都存在且在时如果do,)(lim,)(lim00BxgAxfxxxx==®®设)1(),(0,0,0101xfAxx<-<-<>$>"edde时有当则)2(.)(0,0202edd+<<-<>$Bxgxx时有当于是有同时成立与不等式时则当令，xgxfxx)2(),1()()(,0},,,min{021'£<-<=ddddd,)()(ee+<£<-BxgxfA.,2BABA£+<的任意性知由从而ee4.保不等式性E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

如果函数f(x)、g(x)及h(x)满足下列条件

(1)g(x)

f(x)

h(x)

(2)limg(x)

A

limh(x)

A

那么limf(x)存在

且limf(x)

A

证明),(0,0,0101xgAxx，<-<-<>$>"edde时有当按假设.)(0,0202edd+<<-<>$Axhxx时有当故有同时成立时上两不等式与则当令,)()()(0},,min{021xhxfxgxx££<-<=dddd,)()()(ee+<££<-AxhxfxgA.)(lim)(0Axf，Axfxx=<-®即由此得e5.迫敛性E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410(2)limf(x)

g(x)=limf(x)

limg(x)=A

B

推论1

如果limf(x)存在

而c为常数

则lim[c

f(x)]=c

limf(x)

推论2如果limf(x)存在

而n是正整数

则lim[f(x)]n=[limf(x)]n

如果limf(x)=A

limg(x)=B

那么6.极限的四则运算法则(1)lim[f(x)

g(x)]=limf(x)

limg(x)=A

B

E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604107函数极限与数列极限的关系

如果当x

x0时f(x)的极限存在

{xn}为f(x)的定义域内任一收敛于x0的数列

且满足xn

x0(n

N

)

那么相应的函数值数列{f(xn)}必收敛

且E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604108.子列收敛性(函数极限与数列极限的关系)定义定理E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例如,函数极限与数列极限的关系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.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例7证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410二者不相等,E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410求极限举例讨论

提示

例1

解

>>>

例2

解

E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

解

例3

解

例4

根据无穷大与无穷小的关系得因为E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410讨论

提示

当Q(x0)

P(x0)

0时

约去分子分母的公因式(x

x0)

E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410先用x3去除分子及分母

然后取极限

解

先用x3去除分子及分母

然后取极限

例5

解:

例6

E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410讨论提示

例7

解

所以E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

解

当x

时

分子及分母的极限都不存在

故关于商的极限的运算法则不能应用

例8

是无穷小与有界函数的乘积

E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410(1),唯一性;

作业

小结(2),局部有界性;(3),局部保号性;(4),保不等式性;(5),迫敛性;P47:1,2,3,5,6,7,8,9.(6),四则运算法则;(7),函数极限与数列极限的关系;(8),复合函数的四则运算法则.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410第三章函数极限§3

函数极限存在的条件E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、极限存在准则1.夹逼准则证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410上两式同时成立,上述数列极限存在的准则可以推广到函数的极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410注意:准则

和准则'称为夹逼准则.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例1解由夹逼定理得E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604102.单调有界准则单调增加单调减少单调数列几何解释:E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例2证(舍去)E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410第三章函数极限§4

两个重要极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410二、两个重要极限(1)E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410注:

这是因为

令u=a(x)

则u

0

于是第一个重要极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

例1

解

解

例2

E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例3注：在上例中，应用公式（14—1）时，我们使用了代换,在运算熟练后可不必代换，直接计算：E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例4.求极限:E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例5.求极限:

E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

练习1.求下列极限:E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410二.关于极限设有函数，根据下表观察的变化趋势。2.718152.716922.704812.5937410000100010010…..2.718282.71827…1000000100000E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604102.718152.716922.704812.59374-10000-1000-100-10…..2.718282.71827…-1000000-100000时，均趋于一个确定的数2.71828…用e表示该数,e是无理数。e=2.718281828…E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410注意：2.底数中的无穷小量（可以是字母或是代数式）和指数互为倒数。1.公式中底数的极限是1，指数的极限是无穷大,函数极限为型E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410定义第二个重要极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410类似地,E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例6,求极限

解：E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例7解：E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例8解：E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例9解例10解E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

练习2.求下列极限:E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410练习E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410小结：（1）分子、分母含有三角函数且在自变量指定的变化趋势下是“”型。（2）公式中的“”可以是趋向于零的代数式。（3）注意三角函数有关公式的应用。（1）函数在自变量指定的变化趋势下是“”型。（2）应用公式解题时，注意将底数写成1与一个无穷小量的代数和的形式，该无穷小量与指数互为倒数。（3）注意求极限过程中运用指数的运算法则。E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410作业:P58:1(1)~(10),2(1)~(6),3,4(1)~(2).E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410三、小结1.两个准则2.两个重要极限夹逼准则;单调有界准则.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410思考题求极限E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410思考题解答E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、填空题:练习题E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410二、求下列各极限:E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410练习题答案E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410第三章函数极限§5

无穷小量与无穷大量E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410则称f(x)是该极限过程中的一个无穷小量(省去xxo,x

的极限符号“lim”表示任一极限过程).定义1.

若limf(x)=0,一、无穷小E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410一、无穷小1、定义:极限为零的变量称为无穷小.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例如,注意（1）无穷小是变量,不能与很小的数混淆;E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410注2：无穷小量与极限过程分不开,不能脱离极限过程谈无穷小量,小量,但如sinx是x0时的无穷E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410注3：由于limC=C(常数),注4：0是任何极限过程的无穷小量.所以,除0外的任何常数不是无穷小量.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C243604102、无穷小与函数极限的关系:证必要性充分性E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410意义（1）将一般极限问题转化为特殊极限问题(无穷小);3、无穷小的运算性质:定理2在同一过程中,有限个无穷小的代数和仍是无穷小.证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410注意

无穷多个无穷小的代数和未必是无穷小.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410定理3有界函数与无穷小的乘积是无穷小.证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410推论1在同一过程中,有极限的变量与无穷小的乘积是无穷小.推论2常数与无穷小的乘积是无穷小.推论3有限个无穷小的乘积也是无穷小.都是无穷小E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410二、无穷大绝对值无限增大的变量称为无穷大.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410二、无穷大量定义2：若

>0(无论多么大),

记作：

>0(或

X>0),当0<|x–xo|<

(或|x|>X)时，有|f(x)|>M,则称f(x)是x

x0(或x)时的无穷大量.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410若以“f(x)>M”代替定义中的“|f(x)|>M”,就得到正无穷大量的定义.若以“f(x)<–M”代替定义中的“|f(x)|>M”,就得到负无穷大量的定义.分别记作：

>0,

>0(或

X>0),当0<|x–xo|<

(或|x|>X)时，有|f(x)|>M,E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410特殊情形：正无穷大，负无穷大．注意（1）无穷大是变量,不能与很大的数混淆;（3）无穷大是一种特殊的无界变量,但是无界变量未必是无穷大.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410不是无穷大．无界，E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410例2:试从函数图形判断下列极限.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410解：(1)xy0xyy=tgxxyE6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410(2)xoyxxyyx+x–E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410注1：若在定义2中，将“f(x)”换成“xn”,注2：若limf(x)=,将“X”换成“N”,将“x”换成就得到数列xn为无穷大量定义.“n”,则表示在该极限过程中f(x)的极限不存在.

>0,X>0,当|x|>X时,有|f(x)|>M,E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410注3：不能脱离极限过程谈无穷大量.注4：无穷大量一定是无界量,任何常量都不是无穷大量.但无界量不一定是无穷大量.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410说明

>0,x0(–,+)，使得|x0sinx0|>M即可.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410三、无穷小与无穷大的关系定理4在同一过程中,无穷大的倒数为无穷小;恒不为零的无穷小的倒数为无穷大.证E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410意义

关于无穷大的讨论,都可归结为关于无穷小的讨论.E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

证明

设

及

是当x

x0时的两个无穷小

则

0

1

0

当0

|x

x0|

1

时

有|

|

2

0

当0

|x

x0|

2

时

有|

|

取

min{

1

2}

则当0

|x

x0|

时

有这说明

也是当x

x0时的无穷小

|

|

|

|

|

|

2

定理1

有限个无穷小的和也是无穷小

仅就两个x

x0时的无穷小情形证明

举例:

当x

0时

x与sinx都是无穷小

所以x

sinx也是当x

0时的无穷小

四、无穷小的性质E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410

设函数u在x0的某一去心邻域{x|0

|x

x0|

1}内有界

即

M

0

使当0

|x

x0|

1时

有|u|

M

又设

是当x

x0时的无穷小

即

0

存在

2

0

使当0

|x

x0|

2时

有|

|

取

min{

1

2}

则当0

|x

x0|

时

有|u

|

|u|

|

|

M

这说明u

也是当x

x0时的无穷小

证明

定理2有界函数与无穷小的乘积是无穷小

定理1

有限个无穷小的和也是无穷小

四、无穷小的性质E6636B02012BD195C019CE06C16E3084C53D00E32711A616D0819A133DEA58FE8FF97F03BC41BA96FB4BDEF0472B73A72FC0F2BE2AA53FFE6A09217957F6D67DBF2DB6715F2B3FB878C24360410举例:推论2

有限个无穷小的乘积也是无穷小

定理2有界函数与无穷小的乘积是无穷小

定理1

有限个无穷小