利用等价无穷小在分子或者分母中对元素做整体替换

一、题目

$$I=\underset{x\to 0}{lim}\frac{\arctan x–\tan x}{\sin x–\sin (\sin x)}=?$$

难度评级：

二、解析

$$\begin{array}{rl}I=& \underset{x\to 0}{lim}\frac{\arctan x–\tan x}{\sin x–\sin (\sin x)}\\ =& \underset{x\to 0}{lim}\frac{\arctan x–\tan x}{x–\sin (x)}\\ =& \underset{x\to 0}{lim}\frac{(\arctan x–x)–(\tan x–x)}{x–\sin x}\\ =& \underset{x\to 0}{lim}\frac{(-\frac{1}{3}{x}^3)–\left(\frac{1}{3}{x}^3\right)}{\frac{1}{6}{x}^3}\\ =& \frac{-\frac{2}{3}}{\frac{1}{6}}=-4.\end{array}$$

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