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高等数学

2019-05-29 09:52:25   4  举报

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高等数学的初步知识总结

高等数学

高数；数学

大学

学校教育

作者其他创作

大纲/内容

第一章函数与极限

希腊字母表

第一节映射与函数

集合的概念和运算，区间和邻域

映射又称为算子 非空集X到数集Y的映射，称为X上的泛函 非空集X到它自身的映射，称为X上的变换 实数集X到实数集Y的映射，称为定义在X上的函数

逆映射与复合映射 满射、单射、一一映射（双射） 只有单射才存在逆映射

函数的有界性、单调性、奇偶性、周期性

反函数与复合函数 基本初等函数：幂函数、指数函数、对数函数、三角函数、反三角函数 由常数和基本初等函数经过有限次的四则运算和有限次的函数复合步骤所构成并可以用一个式子表示的函数，称为初等函数。

第二节数列的极限

在解决实际问题中逐渐形成的这种极限方法，已成为高等数学中的一种基本方法。

收敛数列的性质

定理1（极限的唯一性）如果数列收敛，那么它的极限唯一

定理2（收敛数列的有界性）如果数列收敛，那么数列一定有界

定理3（收敛数列的保号性）

定理4（收敛数列与其子数列间的关系）如果数列收敛于a，那么他的任一子数列也收敛，且极限也是a

第三节函数的极限

函数极限的定义

自变量趋于有限值时函数的极限

自变量趋于无穷大时函数的极限

函数极限的性质

定理1 （函数极限的唯一性）如果函数极限存在，那么这极限唯一

定理2（函数极限的局部有界性）

定理3（函数极限的局部保号性）

定理4 （函数极限与数列极限的关系）若函数收敛于A，则函数定义域内任一收敛于X的数列必收敛且等于A

第四节无穷小与无穷大

无穷小

如果函数当自变量趋近某个值或∞时极限为零，那么称函数为当X趋近某个值或∞时的无穷小

定理1 函数具有极限A的充分必要条件是f(x)=A+a，其中a是无穷小

无穷大

如果函数当自变量趋近某个值或∞时无限增大，那么称函数为当X趋近某个值或∞时的无穷大

定理2 在自然定义域内，无穷大函数的倒数是无穷小；无穷小的函数的倒数是无穷大

第五节极限运算法则

定理1 有限个无穷小的和也是无穷小

定理2 有界函数与无穷小的乘积是无穷小

推论1 常数与无穷小的乘积是无穷小

推论2 有限个无穷小的乘积也是无穷小

定理3 有限个极限为常数的函数，满足四则运算法则

推论1 求极限时，常数因子可以提到极限记号外面

推论2 求极限时，指数为常数，也可提到极限记号外面

定理4 数列的极限也符合四则运算法则

定理5 函数f1≥函数f2，且limf1=A，limf2=B，则A≥B

定理6 （复合函数的极限运算法则）

第六节极限存在准则两个重要极限

准则1：夹逼准则：函数f1≥函数f2≥函数f3，且limf1=A，limf3=A，则limf2=A

lim（sinx/x），当x→0时的极限为1

准则2：单调有界数列必有极限（充分而不必要）（收敛数列不一定单调）

左邻域内单调并且有界，则左极限必定存在

lim（1+1/x）^x，当x→∞时的极限存在，为e

柯西（Cauchy）极限存在准则：（几何意义）对于任意给定的整数ε，在数轴上一切具有足够大号码的点Xn中，任意两点间的距离小于ε

第七节无穷小的比较

高阶β=o(α)、低阶、同阶、k阶、等价（α～β）无穷小

定理1 β与α是等价无穷小的充要条件为：β=α+o(α)

定理2 求两个无穷小之比的极限时，分子及分母都可以用等价无穷小来代替

第八节函数的连续性与间断点

连续函数的图形是一条连续而不间断的曲线

函数f(x)在点Xo的某去心邻域内有定义，在此前提下，如果函数f(x)有下列三种情形之一： （1）在X=Xo没有定义； （2）虽在X=Xo有定义，但limf(x)在X→Xo不存在； （3）虽在X=Xo有定义，且limf(x)在X→Xo存在，但limf(x)≠f(Xo)； 则函数f(x)在点Xo为不连续，而点Xo称为函数f(x)的不连续点或间断点。

间断点分成两类： （1）左右极限都存在，称为第一类间断点； 特别的，左右极限存在并且相等，称为可去间断点，不相等者称为跳跃间断点。 （2）不是第一类间断点的任何间断点，称为第二类间断点； 例如无穷间断点、震荡间断点

第九节连续函数的运算与初等函数的连续性

定理1 连续函数的和、差、积、商，仍然连续

定理2 单调连续函数的反函数也单调连续

定理3 复合函数的值域内外传递法则

定理4 复合函数的连续性

初等函数的连续性： 基本初等函数在它们的定义域内都是连续的； 一切初等函数在其定义区间内都是连续的

第十节闭区间上连续函数的性质

定理1 （有界性与最大值最小值定理）在闭区间上连续的函数在该区间上有界且一定能取得他的最大值和最小值

定理2 （零点定理）设函数f(x)在闭区间[a，b]上连续，且f(a)与f(b)异号（即f(a)f(b)<0），那么在开区间（a，b）内至少有一点ξ，使f(ξ)=0

定理3 （介值定理）设函数f(x)在闭区间[a，b]上连续，且在这区间的端点取不同的函数值f(a)=A及f(b)=B 那么，对于A与B之间的任意一个数C，在开区间（a，b）内至少有一点ξ，使得f(ξ)=C（a<ξ<b）

定理3推论在闭区间上连续的函数必取得介于最大值M和最小值m之间的任何值

定理4 （一致连续性）如果函数f(x)在闭区间[a，b]上连续，那么它在该区间上一致连续

第二章导数与微分

第一节导数概念

引例： 1.直线运动的速度（瞬时速度） 2.切线问题（切线的斜率）

在某处，Δy/Δx（Δx→0）的极限存在，则函数在某处可导、具有导数、导数存在； 若在开区间内每点处都可导，则构成一个新函数-称为原来函数的导函数。

导数的几何意义--切线方程

函数可导性与连续性的关系 函数在某处可导，则函数在该点必连续； 函数在某处连续，但函数在该点不一定可导。

第二节函数的求导法则

函数的和、差、积、商的求导法则

定理1 若函数<img width="69" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D69/sign=8d7adf6716950a7b71354dcd0bd18988/8c1001e93901213f797cca8d5fe736d12f2e9510.jpg">都可导，则经过其四则运算后的复合函数在其自然定义域内可导。 <img width="206" height="23" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: 微软雅黑; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: center; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D206/sign=cc4b795990504fc2a65fb705d3dce7f0/810a19d8bc3eb135d09f42eaad1ea8d3fd1f4403.jpg"> <img width="290" height="23" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: 微软雅黑; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D290/sign=128cab02fd246b607f0eb57ddbf91a35/77094b36acaf2edd0f49bcb8861001e939019323.jpg"> <img width="234" height="49" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: 微软雅黑; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D234/sign=e49e03afc8fdfc03e178e4bbe03e87a9/aa64034f78f0f73683a1b9500155b319ebc41307.jpg"> <img width="406" height="23" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D406/sign=5b565b3144c2d562f608d1edd11090f3/6d81800a19d8bc3e535bf39b898ba61ea8d34514.jpg"> <img width="116" height="23" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D116/sign=c2d507bf9782d158bf825db0b60b19d5/d53f8794a4c27d1ec1c7ba9510d5ad6edcc438f9.jpg"> <img width="367" height="58" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D367/sign=eaf7039cf7039245a5b5e709b095a4a8/a50f4bfbfbedab642a76b891fc36afc379311e5e.jpg">

反函数的求导法则

定理2 若函数<img width="60" height="19" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D60/sign=0acc26429cdda144de096fb2b2b7de29/cc11728b4710b912cee774afc8fdfc03924522b8.jpg">严格单调且可导，则其反函数<img width="59" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D59/sign=ab0ae7bb8acb39dbc5c0675fd1167065/3c6d55fbb2fb43161becf92b2ba4462308f7d3f4.jpg">的导数存在且<img width="97" height="41" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D97/sign=c7bb4c8ca064034f0bcdce01aec355d5/eaf81a4c510fd9f95b6ad7382e2dd42a2834a466.jpg">

复合函数求导法则

若<img width="58" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D58/sign=3cb07e5c7bcf3bc7ec00cde4d0006a53/1e30e924b899a90117b1d96716950a7b0308f5df.jpg">在点x可导，<img width="60" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D60/sign=732e13799c2bd40746c7d0fd7b89491a/21a4462309f79052e228ca6f07f3d7ca7bcbd5a8.jpg">在相应的点u也可导，则其复合函数<img width="81" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D81/sign=85744bbfbf8f8c54e7d3c82e3a293736/c995d143ad4bd11359d53a6351afa40f4bfb0542.jpg">在点x可导且<img width="140" height="22" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D140/sign=66a3c3aa0146f21fcd345a57c6256b31/0df431adcbef7609fa5f14ce25dda3cc7cd99e0f.jpg">。

特殊求导法则

对数求导法则

函数<img width="103" height="22" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D103/sign=7a90396b12d8bc3ec20802cab18ba6c8/dcc451da81cb39dba0eceb07db160924ab1830b6.jpg">被称为幂指函数，在经济活动中会大量涉及此类函数。 注意到它很特别，既不是指数函数又不是幂函数，它的幂底和指数上都有自变量x，所以不能用初等函数的微分法处理了。 这里介绍一个专门解决此类函数的方法，对数求导法。 对于<img width="103" height="22" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D103/sign=7a90396b12d8bc3ec20802cab18ba6c8/dcc451da81cb39dba0eceb07db160924ab1830b6.jpg">两边取对数（当然取以为e底的自然对数计算更方便）。 由对数的运算性质：<img width="163" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D163/sign=ea5cddf4017b020808c93be751d8f25f/1b4c510fd9f9d72ae403e54cdf2a2834349bbb78.jpg"> 再对两边求导： <img width="295" height="40" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D295/sign=d33f9322f0f2b211e02e8247ff806511/77c6a7efce1b9d163efd6f30f8deb48f8c5464ad.jpg"> <img width="317" height="43" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: center; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D317/sign=1d5a3d47c4fc1e17f9bf8a307d91f67c/63d9f2d3572c11dff71f3572682762d0f603c291.jpg">

参数表达函数的求导法则

若参数表达<img width="46" height="36" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D46/sign=3f781fad77d98d1072d40d37203fd0b9/1ad5ad6eddc451da906e85babdfd5266d11632c2.jpg">为一个y关于x的函数，<img width="58" height="19" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D58/sign=2b34da68c23d70cf48faaa05f8dc6b22/29381f30e924b899fd0bf16965061d950a7bf6a8.jpg">必存在反函数<img width="62" height="23" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D62/sign=2ea7924fb2389b503cffe35084352684/d1160924ab18972bcc71889bedcd7b899e510a13.jpg">，于是代入<img width="145" height="27" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D145/sign=a5e78e9b898ba61edbeecc2b743597cc/80cb39dbb6fd5266c4c4b445a018972bd50736f7.jpg">  这便是y通过中间变量t的关于x的函数的抽象表达，（实际中未必能写出t关于x的反函数式子，也没必要这样做）。 利用反函数求导法则和复合函数求导法则，可得 <img width="187" height="53" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D187/sign=e53b4db25c2c11dfdad1bb2b54266255/d8f9d72a6059252d53c897553f9b033b5ab5b980.jpg"> 这便是参数方程表达的y关于x的函数的求导公式。

隐函数求导法则

若<img width="75" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D75/sign=02de5b9f9225bc312f5d039d5edf303c/4afbfbedab64034f57db4c57a4c379310a551d48.jpg">中存在隐函数<img width="58" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D58/sign=f4ce0b5438fa828bd5239debfc1f1a04/b151f8198618367a9244f08c25738bd4b21ce596.jpg">，这里仅是说y为一个x的函数并非说y一定被反解出来为显式表达。 即<img width="94" height="19" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D94/sign=b6690750576034a82de2b485ca130edc/4afbfbedab64034f57bc4c57a4c379310a551d69.jpg">，尽管y未反解出来，只要y关于x的隐函数存在且可导，我们利用复合函数求导法则仍可以求出导数。

基本初等函数的导数表

第三节高阶导数

二阶或二阶以上的导数统称高阶导数

排列组合公式 <img style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial; font-size: 13px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; max-width: 498px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="formula" src="https://ss1.baidu.com/6ONXsjip0QIZ8tyhnq/it/u=1488885628,2099751177&fm=58"> <img style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial; font-size: 13px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; max-width: 498px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="formula" src="https://ss1.baidu.com/6ONXsjip0QIZ8tyhnq/it/u=2718301618,521244&fm=58">  公式中A(n,m)为排列数公式，C(n,m)为组合数公式。

莱布尼茨公式 <img width="175" height="49" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D175/sign=c026e6c6a718972ba73a04cdd3cc7b9d/3812b31bb051f8199628d998d6b44aed2f73e7d7.jpg">

第四节隐函数及由参数方程所确定的函数的导数            相关变化率

隐函数、参数方程的导数，参考第二节“特殊求导法则”

设x=x(t)及y=y(t)都是可导函数，而变量x与y之间存在某种关系，从而变化率dx/dt与dy/dt间也存在一定关系。 这两个相互依赖的变化率称为相关变化率。 相关变化率问题就是研究这两个变化率之间的关系，以便从其中一个变化率求出另一个变化率。

第五节函数的微分

微分的定义

由函数B=f(A)，得到A、B两个数集，在A中当dx靠近自己时，函数在dx处的极限叫作函数在dx处的微分。 微分的中心思想是无穷分割。 微分是函数改变量的线性主要部分。 通常把自变量x的增量 Δx称为自变量的微分，记作dx，即dx = Δx。于是函数y = f(x)的微分又可记作dy = f'(x)dx。 函数因变量的微分与自变量的微分之商等于该函数的导数。因此，导数也叫做微商。 当自变量X改变为X+△X时，相应地函数值由f(X)改变为f(X+△X)，如果存在一个与△X无关的常数A， 使f(X+△X)-f(X)和A·△X之差是△X→0关于△X的高阶无穷小量，则称A·△X是f(X)在X的微分，记为dy，并称f(X)在X可微。

微分的几何意义

微分概念是在解决直与曲的矛盾中产生的，在微小局部可以用直线去近似替代曲线，它的直接应用就是函数的线性化。

基本初等函数的微分公式与微分运算法则

微分在近似计算中的应用

函数的近似计算

误差估计 绝对误差限、相对误差限

第三章微分中值定理与导数的应用

第一节微分中值定理

费马引理 函数f(x)在点ξ的某邻域U（ξ)内有定义，并且在ξ处可导，如果对于任意的x∈U（ξ)，都有f(x)≤f(ξ) (或f(x)≥f(ξ) )，那么f '(ξ）=0。 该定理通过证明函数的每一个极值都是驻点（函数的导数在该点为零），给出了一个求出可微函数的最大值和最小值的方法。 需要注意的是，费马引理仅仅给出了函数在某个点为极值的必要条件。也就是说，有些驻点可以不是极值，它们是拐点。要想知道一个驻点是不是极值，并进一步区分极大值和极小值，我们需要分析二阶导数（如果它存在）。当该点的二阶导数大于零时，该点为极小值点；当该点的二阶导数小于零时，该点为极大值点。若二阶导数为零，则无法用该法判断，需列表判断。）。

罗尔定理 如果函数f(x)满足： 在闭区间[a,b]上连续； 在开区间(a,b)内可导； 在区间端点处的函数值相等，即f(a)=f(b)， 那么在(a,b)内至少有一点ξ(a<ξ<b)，使得 f'(ξ)=0. 几何上，罗尔定理的条件表示，曲线弧（方程为）是一条连续的曲线弧，除端点外处处有不垂直于x轴的切线，且两端点的纵坐标相等。 而定理结论表明：弧上至少有一点，曲线在该点切线是水平的。

拉格朗日定理 如果函数 f(x) 满足： 1)在闭区间[a,b]上连续； 2)在开区间(a,b)内可导。 那么：在(a,b)内至少有一点ξ(a<ξ<b)， 使等式 f(b)-f(a)=f′(ξ)(b-a) 成立。 拉格朗日中值定理的几何意义是：曲线上必然存在至少一点，过该点的切线的斜率和连接曲线（a，b）的割线的斜率相同； 或者说，曲线上必然存在至少一点可以做割线（a，b）的平行线

记 <img width="183" height="19" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D183/sign=d57f12b2c7fc1e17f9bf88397991f67c/622762d0f703918f363a986e593d269758eec4af.jpg"> ,令<img width="114" height="16" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D114/sign=5d4d33aab93533faf1b6972f9cd2fdca/50da81cb39dbb6fd16f35f820124ab18962b37ed.jpg">  ,则有 <img width="364" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D364/sign=718cbfe29025bc312f5d079e6ade8de7/9e3df8dcd100baa191682d974e10b912c8fc2e51.jpg"> 上式称为有限增量公式。 我们知道函数的微分<img width="93" height="19" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss1.bdstatic.com/-vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D93/sign=1f3085a569d0f703e2b299df09fa9dd4/b8389b504fc2d5621016035fec1190ef76c66c62.jpg">是函数的增量Δy的近似表达式，一般情况下只有当|Δx|很小的时候，dy和Δy之间的近似度才会提高；而有限增量公式却给出了当自变量x取得有限增量Δx（|Δx|不一定很小）时，函数增量Δy的准确表达式，这就是该公式的价值所在。

柯西定理 如果函数f(x)及F(x)满足 (1)在闭区间[a,b]上连续； (2)在开区间(a,b)内可导； (3)对任一x∈(a,b)，F'(x)≠0 那么在(a,b) 内至少有一点ξ，使等式 [f(b)-f(a)]/[F(b)-F(a)]=f'(ξ)/F'(ξ)成立

第二节洛必达法则

未定式 两个无穷小之比或两个无穷大之比的极限可能存在，也可能不存在。 因此，求这类极限时往往需要适当的变形，转化成可利用极限运算法则或重要极限的形式进行计算。

以当x→x0时为例，如果符合上述条件的函数f(x)与g(x)都在x0的邻域内存在n阶导数，那么： <img width="398" height="45" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D398/sign=5b46b3f37b1ed21b7dc928ec956fddae/79f0f736afc37931d687a428edc4b74542a911c7.jpg"> 这就是洛必达法则。

零比零型 <img width="10" height="35" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D35/sign=36468b15faedab6470724bc5f6365669/6159252dd42a2834a6d2c3d55ab5c9ea14cebffc.jpg"> 若函数<img width="32" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D32/sign=7449c9ee9f510fb37c197195d8334498/7acb0a46f21fbe09f9e744436a600c338744ad28.jpg">和<img width="30" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D30/sign=b1595de1b0fb43161e1f7c7a20a4937f/8d5494eef01f3a29a1c1e8e49b25bc315c607c8a.jpg">满足下列条件： ⑴<img width="87" height="24" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D87/sign=f5b465afa686c9170c035f3ec83d6f99/b21c8701a18b87d647012a48060828381f30fd22.jpg"> ，<img width="85" height="24" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D85/sign=de0c04cd932397ddd27995015982df02/f9dcd100baa1cd11c9bf4f0abb12c8fcc3ce2d8a.jpg">  ； ⑵ 在点<img width="8" height="8" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D8/sign=31c216ab7dd98d1072d401012177f2/7c1ed21b0ef41bd5b27168a350da81cb39db3d42.jpg">的某去心邻域内两者都可导，且<img width="62" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D62/sign=64609cf4f21f3a295ec8d6cc9925347d/83025aafa40f4bfb13d9a37e014f78f0f736188a.jpg"> ； ⑶ <img width="98" height="40" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D98/sign=9fffd9d2a586c9170c035e31c93d6f07/8ad4b31c8701a18b8de32f659c2f07082838fe8b.jpg"> （ <img width="12" height="12" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D12/sign=4d8171eaf536afc30a0c3b67b2195294/5ab5c9ea15ce36d39bf7bf1a38f33a87e950b12c.jpg">可为实数，也可为 ±∞ ），则：  <img width="180" height="40" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: center; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D180/sign=c5dc9dd0d3c8a786ba2a4e065708c9c7/0e2442a7d933c8953180a551d31373f082020019.jpg">

无穷比无穷型 <img width="16" height="30" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D30/sign=14d8efc0b8389b503cffe652843526a6/962bd40735fae6cd59805ebf0eb30f2442a70f33.jpg"> 若函数<img width="32" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D32/sign=7449c9ee9f510fb37c197195d8334498/7acb0a46f21fbe09f9e744436a600c338744ad28.jpg">和<img width="30" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D30/sign=afb03cb25cdf8db1b82e7a6408230c9b/0b55b319ebc4b745b0798a3ccdfc1e178a82152e.jpg">满足下列条件： ⑴ <img width="93" height="24" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss1.bdstatic.com/-vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D93/sign=1f26b836b9de9c82a265f58c6d8103ed/b64543a98226cffc9369e3bdb3014a90f603ea7a.jpg"> ，<img width="99" height="25" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D99/sign=7c334b8a1cce36d3a6048f393bf3f19e/32fa828ba61ea8d3da99c3549e0a304e251f5822.jpg">  ； ⑵ 在点<img width="8" height="8" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D8/sign=31c216ab7dd98d1072d401012177f2/7c1ed21b0ef41bd5b27168a350da81cb39db3d42.jpg">的某去心邻域内两者都可导，且<img width="62" height="18" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D62/sign=2edb9fc7a344ad342abf8485d1a2dd41/738b4710b912c8fc7c36cbe7fe039245d78821d6.jpg"> ； ⑶ <img width="104" height="40" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D104/sign=8d6324e6c0177f3e1434f80d44ce3bb9/9f510fb30f2442a785984bf5d843ad4bd01302cf.jpg">（<img width="12" height="12" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D12/sign=4d8171eaf536afc30a0c3b67b2195294/5ab5c9ea15ce36d39bf7bf1a38f33a87e950b12c.jpg">可为实数，也可为<img width="25" height="9" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/94o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D25/sign=fd7c465022a446237acaa2679822ef28/e1fe9925bc315c60a4f657c58fb1cb13495477bc.jpg"> ），则： <img width="194" height="40" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: center; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D194/sign=7e01266989d4b31cf43c90b2b3d7276f/472309f79052982246e204e9d5ca7bcb0b46d4c2.jpg">

注意： 该定理所有条件中，对<img width="48" height="8" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D48/sign=6d94c3ee9f510fb37c19769fd83344f5/d6ca7bcb0a46f21f3550d304f7246b600c33ae05.jpg">的情况，结论依然成立。 不定式极限还有<img width="33" height="12" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/-fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D33/sign=b7633be0808ba61edbeece2c41341923/377adab44aed2e738d82bb068501a18b87d6fab4.jpg">，<img width="19" height="13" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/-Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D19/sign=d1a966f0a9d3fd1f3209a633314e1744/8ad4b31c8701a18bd4f4e7919f2f07082938fed5.jpg">，<img width="15" height="16" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss0.bdstatic.com/-4o3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D16/sign=2d283e2538dbb6fd215be1200b244f97/4e4a20a4462309f7eca027cc730e0cf3d6cad625.jpg">，<img width="22" height="16" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss1.bdstatic.com/9vo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D22/sign=270cc7e629381f309a198aaba801b041/2fdda3cc7cd98d10b8caf9ef203fb80e7aec90d6.jpg">，<img width="46" height="7" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D46/sign=d9df3b03aa64034f0bcdc300aec35545/6159252dd42a2834a6bfc3d55ab5c9ea14cebfd7.jpg">等类型。经过简单变换，它们一般均可化为<img width="10" height="35" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D35/sign=36468b15faedab6470724bc5f6365669/6159252dd42a2834a6d2c3d55ab5c9ea14cebffc.jpg">型或<img width="16" height="30" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss2.bdstatic.com/9fo3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D30/sign=14d8efc0b8389b503cffe652843526a6/962bd40735fae6cd59805ebf0eb30f2442a70f33.jpg">型的极限。 不能在数列形式下直接用洛必达法则，因为对于离散变量<img width="50" height="15" title="" align="absmiddle" style="background-color: transparent; border-bottom-color: rgb(51, 51, 51); border-bottom-style: none; border-bottom-width: 0px; border-image-outset: 0; border-image-repeat: stretch; border-image-slice: 100%; border-image-source: none; border-image-width: 1; border-left-color: rgb(51, 51, 51); border-left-style: none; border-left-width: 0px; border-right-color: rgb(51, 51, 51); border-right-style: none; border-right-width: 0px; border-top-color: rgb(51, 51, 51); border-top-style: none; border-top-width: 0px; color: rgb(51, 51, 51); font-family: arial,&quot;宋体&quot;,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 28px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;" alt="" src="https://gss3.bdstatic.com/7Po3dSag_xI4khGkpoWK1HF6hhy/baike/s%3D50/sign=99e9dd23b4003af349badc60352ace01/314e251f95cad1c808bcc10d7d3e6709c93d5190.jpg">是无法求导数的。但此时有形式类近的斯托尔兹－切萨罗定理（Stolz－Cesàro theorem）作为替代。

第三节泰勒公式