极限

等价无穷小

$\sin x \sim \arcsin x \sim \tan x \sim \arctan x \sim x$

$\ln {(1+x)} \sim e^x - 1 \sim \sqrt{1 + x} - \sqrt{1 - x}\sim x$

$1 - \cos x \sim \sec x - 1 \sim x - \ln {(1 + x)} \sim \frac{1}{2}x^2$

$1-\cos ^{a} x \sim \frac{a}{2} x^{2}$

$(1 + x)^a -1 \sim ax$

$a^x - 1 \sim xlna$

$\log a ^{(1 + x)} \sim \frac{x}{lna}$

$\tan x - x \sim x - \arctan x \sim \frac{x^3}{3}$

$x - \sin x \sim \arcsin x - x \sim \frac{x^3}{6}$

泰勒公式

$e^x = 1 + x + \frac{x^2}{2!} + ··· + \frac{x^n}{n!} +o(x^n)$

$\frac{1}{1-x}=1 + x + x^2 + ...+ x^n + o(x^n)$

$\frac{1}{1+x}=1 - x + x^2 + ...+ (-1)^nx^n + o(x^n)$

$(1 + x)^\alpha = 1 + \alpha x + \frac{\alpha (\alpha - 1)}{2!}x^2 + ··· + \frac{\alpha (\alpha - 1) ··· (\alpha - n + 1)}{n!}x^n + o(x^n)$

$\ln(1 + x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - ··· + (-1)^{n }\frac{1}{n+1}x^{n+1} + o(x^{n+1})$

$\cos x = 1 - \frac{1}{2!}x^2 + \frac{1}{4!}x^4 + o(x^{4})$

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + o(x^{5})$

$\tan x = x + \frac{x^3}{3} + \frac{2}{15} x^5 + o(x^5)$

$\arctan x = x - \frac{x^3}{3} + \frac{x^5}{5}+ o(x^{5})$

$\arcsin x = x +\frac{1}{6} x^{3}+\frac{3}{40} x^{5} + o(x^{5})$

$\sec x = 1+\frac{1}{2} x^{2}+\frac{5}{24} x^{4}+o(x^{4})$

$\csc x = \frac{1}{x}+\frac{1}{6} x+\frac{7}{360} x^{3}+o(x^3)$

$\cot x = \frac{1}{x}-\frac{1}{3} x-\frac{1}{45} x^{3}+o(x^3)$

其他

当 $x>0$ 时, $\ln (1+x)<x ; \quad \sin x<x$

$\lim \atop {x \rightarrow 0}$$\frac{\sin x}{x}=1$

$\lim\atop {x \rightarrow 0}$$(1+x)^{\frac{1}{x}}=\mathrm{e}$

$\lim \atop {x \rightarrow \infty}$$\left(1+\frac{1}{x}\right)^{x}=\mathrm{e}$

$\lim \atop {x \rightarrow 0}$$\frac{a^{x}-1}{x}=\ln a$

$\lim \atop {x \rightarrow \infty}$$\sqrt[n]{n}=1$

$\lim \atop {x \rightarrow \infty}$$\sqrt[n]{a}=1,(a>0)$

$\lim\atop {x \rightarrow \infty}$$\frac{a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}}{b_{m} x^{m}+b_{n-1} x^{m-1}+\cdots+b_{1} x+b_{0}}=\left\{\begin{array}{ll}\frac{a_{n}}{b_{m}}, & n=m \\ 0, & n<m \\ \infty, & n>m\end{array}\right.$

$x \rightarrow+\infty$ 时
$\ln ^{\alpha} x \ll x^{\beta} \ll a^{x}, \text { 其中 } \alpha>0, \beta>0, a>1$
$x \rightarrow \infty$时
$\ln ^{\alpha} n \ll n^{\beta} \ll a^{n} \ll n ! \ll n^{n}, 其中\alpha>0, \beta>0, a>1$

$\lim \atop {n \rightarrow \infty}$ $\sqrt[n]{a_{1}^{n}+a_{2}^{n}+\cdots+a_{m}^{n}}=a$