高等数学-基础概念-函数与极限习题

高等数学-基础概念-函数与极限习题

首先判断未定式类型，使用对应的方法 无法根据未定式求解时，考虑根据定义来做，或者夹逼定理，泰勒展开等 对于含有绝对值的式子求极限，需要分类讨论

根据未定式的形式求极限

\(\frac{0}{0}\)型、\(\frac{\infty}{\infty}\)型

等价无穷小求极限

例1

\(\lim _{x \rightarrow 0} \frac{e^{x}-e^{\ln (1+x)}}{x \arcsin 2 x}\)=？

答：

\(\frac{e^{x}-e^{\ln (1+x)}}{x+0} \frac{e^{x}}{x \arcsin 2 x}=\lim _{x \rightarrow 0} e^{\ln (1+x)} \cdot \frac{e^{x-\ln (1+x)}-1}{2 x^{2}}=\lim _{x \rightarrow 0} \frac{e^{x-\ln (1+x)}-1}{2 x^{2}}\) \(=\lim _{x \rightarrow 0} \frac{x-\ln (1+x)}{2 x^{2}}=\lim _{x \rightarrow 0} \frac{1-\frac{1}{1+x}}{4 x}=\frac{1}{4}\)

例2
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例3
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例4
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例5
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例6
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例7
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这里使用等价无穷小，也可以使用和差化积+泰勒展开

例8
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例9
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根式有理化或因式分解求极限

目的是尽可能的消去0或无穷因子，再用极限运算法则或连续函数极限的求法。

例1
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例2
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例3
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例3
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例4
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例5

方法1根式有理化，方法2等价无穷小替换

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例6
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例7
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洛必达法则求极限

例1
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变量替换求极限

使用洛必达法则运算后，若式子反而比原来更复杂，应想到用变量替换。

例1
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例2
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例3

本题用等价无穷小+变量替换

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直接凑成或者添加因子凑成两个极限的积或和

例1
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例2
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解此题还有一种方法见泰勒公式求极限的例1

例3
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例4
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例5
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此题也可对sin部分直接做等价无穷小代换

例6
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例7
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这道题先凑成两个极限的积，然后多次使用等价无穷小替换

例8

凑因子+洛必达+分子有理化

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例9

凑因子+等价无穷小

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例10
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注意这里分母凑成两项的和，原式的倒数可以拆成两项使用等价无穷小替换，所以这里的分母也可以使用等价无穷小替换

例11
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这道题是凑因子+泰勒展开

提因子留常数加无穷小量

例1
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分子分母同除以因子求极限

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化为对数形式

例1
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例2

化为对数形式+等价无穷小替换

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\(\infty - \infty\)型求极限

通分求极限

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提因子变量替换求极限

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\(0\cdot \infty\)型求极限

变量替换

例1
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\(1^\infty\)型、\(0^0\)型、\(\infty^0\)型求极限

凑自然对数e求极限

例1
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例2
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例3
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例4
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例5
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例6
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化为对数形式求极限

例1
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例2
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例3
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注意这里将分子化为了对数形式，而没有使用e的定义

例4
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例5
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例6
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例7
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例8
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未定式求极限解不出时

根据相关定义求极限

根据导数定义求极限

例1
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根据定积分定义求极限

（分子齐，分母不齐 ）或者（分母齐，分子不齐）用夹逼定理。

分子分母都齐次用积分。

例1
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例2
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例3
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根据恒等变换求极限

例1
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夹逼定理求极限

（分子齐，分母不齐 ）或者（分母齐，分子不齐）用夹逼定理。

分子分母都齐次用积分。

例1
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例2
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例3
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例4
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例5
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例6
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例7
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泰勒展开求极限

例1
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例2
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例3
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例4
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例5
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例6
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可以使用泰勒展开+等价无穷小替换的方法：

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也可以分母有理化+等价无穷小替换

例6
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例7
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例8
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例9
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例10
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本题也可以对分子使用和差化积公式+等价无穷小替换来做

例11
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例12
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带绝对值式子求极限

例1
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微分中值定理求极限

例1
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根据数列求极限

例1
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已知极限求其他

已知极限求参数

例1
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例2
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例3
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例4
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例4
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例5
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例6
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例7
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例8
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例9
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例10
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例11
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例12
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例13
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此题需要根据x的不同讨论极限值，然后根据连续性来做

已知极限求新极限

主要方法有：

1）想办法从待求极限中凑出已知极限，消去

2）根据已知极限得分子分母无穷小阶数高低关系，可以得到一些值

例1
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例2
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例3
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例4
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比较无穷小阶数高低

泰勒展开后比较是比较方便的

例1
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例2
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例3
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比值求极限判断

例1
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例2
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证明极限存在与否

数列极限存在的证明

证明数列极限的6种方法

数列单调有界，则数列收敛，极限存在

极限为A代入递推式可得A

例1
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例2
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例3
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单调有界必收敛，设极限为A代入递推式可得A

反证极限不存在

例1
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例2
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例3
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极限函数的分段讨论

极限函数连续性的讨论

例1
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间断点类型的判断

例1
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例2
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例3
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例4
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例5
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例6
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例7
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例8
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例9
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