\begin{aligned}
&\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}+\cdots+\frac{x^{2n-1}}{(2n-1)!}+o(x^{2n})
\\
&\cos x=1-\frac{x^2}{2}+\frac{x^4}{4!}+\cdots+\frac{x^{2n}}{(2n)!}+o(x^{2n+1})
\\
&e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^n}{n!}+o(x^{n+1})
\\
&\ln(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\cdots+(-1)^{n+1}\frac{x^n}{n}+o(x^n)
\\
&\frac{1}{1-x}=1+x+x^2+x^3+\cdots+x^n+o(x^{n+1})
\\
&(1+x)^a=1+\frac{a}{1!}x+\frac{a(a-1)}{2!}x^2+\frac{a(a-1)(a-2)}{3!}x^3+\cdots+\frac{a(a-1)\cdots(a-n+1)}{n!}x^n+o(x^{n+1})
\end{aligned}对于每一项我们都展开到1阶（cos展开到2阶）：
\begin{aligned}
\sin x&=x+o(x)
\\
\cos x&=1-\frac{x^2}{2}+o(x^2)
\\
e^x&=1+\frac{x}{1!}+o(x)
\\
\ln(1+x)&=x+o(x)
\\
\frac{1}{1-x}&=1+x+o(x^2)
\\
(1+x)^a&=1+\frac{a}{1!}x+o(x)
\end{aligned}从而得到常见的等价无穷小：
\begin{aligned}
\sin x&\sim x
\\
\cos x&\sim 1-\frac{1}{2}x^2
\\
e^x&\sim 1+{x}
\\
\ln(1+x)&\sim x
\\
\frac{1}{1-x}&\sim1+x
\\
(1+x)^a&\sim 1+ax
\\
(1+\beta x)^\alpha &\sim 1+\alpha\beta x
\\
&\Rightarrow \sqrt[n]{1+x} \sim 1+\frac{1}{n}x
\end{aligned}还有一部分是不太常见的，比如：
\begin{aligned}
a^x-1&\sim x\ln a
\\
\log_{a}{(1+x)}&\sim\frac{x}{\ln a}
\\
x-\sin x &\sim \arcsin x - x \sim \frac{1}{6}x^3
\\
\tan x - x&\sim x -\arctan x \sim \frac{1}{3}x^3
\end{aligned}