常见的等价无穷小
========== 核心 ==========
\(\sin x \sim x\)
\(\tan x \sim x\)
\(\arcsin x \sim x\)
\(\arctan x \sim x\)
\(1 - \cos x \sim \frac{1}{2}x^2\)
\(\ln(1 + x) \sim x\)
\(e^x - 1 \sim x\)
\(a^x - 1 \sim x\ln a \ (a>0,a\neq1)\)
========== 拓展 ==========
\((1 + x)^\alpha - 1 \sim \alpha x \ (\alpha \in \mathbb{R})\)
\(\log_a(1 + x) \sim \frac{x}{\ln a} \ (a>0,a\neq1)\)
\(\sinh x \sim x\)
\(\cosh x - 1 \sim \frac{1}{2}x^2\)
\(\tan x - \sin x \sim \frac{1}{2}x^3\)
\(x - \sin x \sim \frac{1}{6}x^3\)
\(\tan x - x \sim \frac{1}{3}x^3\)
\(\arcsin x - x \sim \frac{1}{6}x^3\)
\(x - \arctan x \sim \frac{1}{3}x^3\)