常见等价无穷小和重要极限

最近更新于 2025-03-26 12:24

常见等价无穷小

当 x\rightarrow0 时

\sin x\sim \arcsin x\sim \tan x \sim \arctan x \sim \ln(1+x) \sim e^x-1 \sim x
a^x-1\sim x\ln a
(1+x)^a-1\sim ax
1-\cos^ax\sim \frac{a}{2}x^2
x-\sin x\sim \frac{1}{6}x^3
x-\arcsin x \sim -\frac{1}{6}x^3
x-\tan x\sim -\frac{1}{3}x^3
x-\arctan x\sim \frac{1}{3}x^3
x-\ln(1+x)\sim\frac{1}{2}x^2
\tan x - \sin x=\frac{1}{2}x^3
\arctan x - \arcsin x=-\frac{1}{2}x^3

重要极限

\lim_{x\to0}\frac{sinx}{x}=1
\lim_{x\to0}(1+x)^\frac{1}{x}=e

\begin{array}{l}
\Rightarrow
\lim_{x\rightarrow0}(1+ax)^\frac{b}{x}=e^{ab} \\
\Rightarrow
\lim_{x\rightarrow\infty}(1+\frac{a}{x})^{bx}=e^{ab} \\
\Rightarrow
若\lim\alpha(x)=0，\lim\beta(x)=\infty，且\lim\alpha(x)\beta(x)=A \\
则有\lim[1+\alpha(x)]^{\beta(x)}=e^A
\end{array}