# 这个无穷小到底有多小？

## 二、解析

### 第一步计算

\begin{aligned} \sin x & = x – \frac{1}{6} x ^{ 3 } + o(x ^{ 3 }) \\ \\ \tan x & = x + \frac{1}{3} x ^{ 3 } + o(x ^{ 3 }) \end{aligned}

\begin{aligned} & \tan (\sin x) – \sin (\tan x) \\ & = \left[ \sin x + \frac{1}{3} \textcolor{yellow}{\sin ^{ 3 } x} \right] – \left[ \tan x – \frac{1}{6} \textcolor{yellow}{\tan ^{ 3 } x } \right] + o (x ^{ 3 }) \\ \\ & = \left[ \sin x + \frac{1}{3} \textcolor{yellow}{ x ^{ 3 } } \right] – \left[ \tan x – \frac{1}{6} \textcolor{yellow}{ x ^{ 3 } } \right] + o (x ^{ 3 }) \\ \\ & = \left[ \left( x – \frac{1}{6} x ^{ 3 } \right) + \frac{1}{3} \textcolor{yellow}{ x ^{ 3 } } \right] – \left[ \left( x + \frac{1}{3} x ^{ 3 } \right) – \frac{1}{6} \textcolor{yellow}{ x ^{ 3 } } \right] + o(x ^{ 3 }) \\ \\ & = x – \frac{1}{6} x ^{ 3 } + \frac{1}{3} \textcolor{yellow}{ x ^{ 3 } } – x – \frac{1}{3} x ^{ 3 } + \frac{1}{6} \textcolor{yellow}{ x ^{ 3 } } + o(x ^{ 3 }) \\ \\ & = \textcolor{green}{\boldsymbol{ o(x ^{ 3 }) }} \end{aligned}

### 第二步计算

\begin{aligned} \sin x & = x – \frac{1}{6} x ^{ 3 } + \textcolor{red}{ \frac{1}{120} x^{5} } + o(x ^{ 5 }) \\ \\ \tan x & = x + \frac{1}{3} x ^{ 3 } + \textcolor{red}{ \frac{2}{15} x ^{ 5 } } + o(x ^{ 5 }) \end{aligned}

\begin{aligned} & \tan (\sin x) – \sin (\tan x) \\ & = \left[ \sin x + \frac{1}{3} \textcolor{yellow}{\sin ^{ 3 } x} + \frac{2}{15} \textcolor{yellow}{ \sin ^{5} x } \right] – \left[ \tan x – \frac{1}{6} \textcolor{yellow}{\tan ^{ 3 } x } + \frac{1}{120} \tan ^{ 5 } x \right] + o (x ^{ 5 }) \\ \\ & = \left[ \sin x + \frac{1}{3} \textcolor{yellow}{ x ^{ 3 } } + \frac{2}{15} \textcolor{yellow}{ x ^{ 5 } } \right] – \left[ \tan x – \frac{1}{6} \textcolor{yellow}{ x ^{ 3 } } + \frac{1}{120} \textcolor{yellow}{ x ^{ 5 } } \right] + o (x ^{ 5 }) \\ \\ & = \left[ \left( x – \frac{1}{6} x ^{ 3 } \right) + \frac{1}{3} \textcolor{yellow}{ x ^{ 3 } } + \frac{2}{15} \textcolor{yellow}{ x ^{ 5 } } \right] – \left[ \left( x + \frac{1}{3} x ^{ 3 } \right) – \frac{1}{6} \textcolor{yellow}{ x ^{ 3 } } + \frac{1}{120} \textcolor{yellow}{ x ^{ 5 } } \right] + o(x ^{ 5 }) \\ \\ & = x – \frac{1}{6} x ^{ 3 } + \frac{1}{3} \textcolor{yellow}{ x ^{ 3 } } + \frac{2}{15} \textcolor{yellow}{ x ^{ 5 } } – x – \frac{1}{3} x ^{ 3 } + \frac{1}{6} \textcolor{yellow}{ x ^{ 3 } } – \frac{1}{120} \textcolor{yellow}{ x ^{ 5 } } + o(x ^{ 5 }) \\ \\ & = \textcolor{green}{\boldsymbol{ \frac{3}{24} x ^{ 5 } + o(x ^{ 5 }) }} \end{aligned}