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

$$
I = \lim_{x \rightarrow 0} \frac{\arctan x – \tan x}{\sin x – \sin(\sin x)} = ?
$$

难度评级：

二、解析

$$
\begin{aligned}
I = & \lim_{x \rightarrow 0} \frac{\arctan x – \tan x}{\textcolor{springgreen}{\sin x} – \sin(\textcolor{springgreen}{\sin x})} \\
= & \lim_{x \rightarrow 0} \frac{\arctan x – \tan x}{\textcolor{springgreen}{x} – \sin(\textcolor{springgreen}{x})} \\
= & \lim_{x \rightarrow 0} \frac{(\arctan x – x) – (\tan x – x)}{x – \sin x} \\
= & \lim_{x \rightarrow 0} \frac{\left( -\frac{1}{3}x^{3} \right) – \left( \frac{1}{3}x^{3} \right)}{\frac{1}{6} x^{3}} \\
= & \frac{-\frac{2}{3}}{\frac{1}{6}} = -4.
\end{aligned}
$$

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