B站求极限例题大赏：洛指展

注意事项

• l(⋯)指代$\underset{x\to 0}{\lim }(\cdots )$，l⁺(⋯)指代$\underset{x\to 0^{+}}{\lim }(\cdots )$；同理，L(⋯)指代$\underset{x\to \infty }{\lim }(\cdots )$，L⁺(⋯)指代$\underset{x\to +\infty }{\lim }(\cdots )$

常用展式

函数	展式（按展式各项从小到大排列）	收敛半径	备注
$e^x$	$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$	$R$
$e^{-x}$	$1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\cdots$	$R$
$\sin x$	$x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$	$R$
$\cos x$	$1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$	$R$
$\tan x$	$x+\frac{x^3}{3}+\frac{2}{15}{x}^5+\cdots$	$R$	$\{\begin{array}{c}{a}_1=\frac{1}{1!}\\ a_3=\frac{a_1}{2!}-\frac{1}{3!}\\ a_5=\frac{a_3}{2!}-\frac{a_1}{4!}+\frac{1}{5!}\\ \cdots \end{array}$
$\cosh x=\frac{e^x+e^{-x}}{2}$	$1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\cdots$	$R$
$\sinh x=\frac{e^x-e^{-x}}{2}$	$x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots$	$R$
$\arctan x$	$x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots$	$[-1,1]$
$\ln (1+x)$	$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots$	$(-1,1]$
$-\ln (1-x)$	$x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+\cdots$	$[-1,1)$
$\begin{array}{c}1\\ \overline{1+x}\end{array}$	$1-x+x^2-x^3+\cdots$	$(-1,1)$
$\begin{array}{c}1\\ \overline{1+x^2}\end{array}$	$1-x^2+x^4-x^6+\cdots$	$(-1,1)$
$\begin{array}{c}x\\ \overline{1+x}\end{array}$	$x-x^2+x^3-x^4+\cdots$	$(-1,1)$
$\begin{array}{c}x\\ \overline{1+x^2}\end{array}$	$x-x^3+x^5-x^7+\cdots$	$(-1,1)$
$\begin{array}{c}1\\ \overline{1-x}\end{array}$	$1+x+x^2+x^3+\cdots$	$(-1,1)$
$\begin{array}{c}1\\ \overline{1-x^2}\end{array}$	$1+x^2+x^4+x^6+\cdots$	$(-1,1)$
$\begin{array}{c}x\\ \overline{1-x}\end{array}$	$x+x^2+x^3+x^4+\cdots$	$(-1,1)$
$\begin{array}{c}x\\ \overline{1-x^2}\end{array}$	$x+x^3+x^5+x^7+\cdots$	$(-1,1)$
$\left|\begin{array}{c}x\end{array}\right|$，但使用傅里叶级数，可用于求自然数的倒数平方和	$\frac{\pi }{2}-\frac{4}{\pi }(\frac{\cos x}{1^2}+\frac{\cos 3x}{3^2}+\frac{\cos 5x}{5^2}+\cdots )$	$[-\pi ,\pi ]$

趋于零

【BV14V411B7eC】
$l^{+}[\frac{1}{x}(\frac{\pi }{4}-\arctan \frac{1}{x+1})]$
$=l[\underset{x\to 0}{\lim }(-\arctan \frac{1}{x+1}{)}^{\prime }]$
$=l[\frac{(x+1{)}^2}{(x+1{)}^2+1}\cdot \frac{1}{(x+1{)}^2}]$
$=\frac{1}{2}$

【BV15L4y1u7zb】
$l[\frac{(e^{\sin x}+\sin x{)}^{\frac{1}{\sin x}}-(e^{\tan x}+\tan x{)}^{\frac{1}{\tan x}}}{x^3}]$
$=l[\frac{e^{\frac{\ln (e^{\sin x}+\sin x)}{\sin x}}-e^{\frac{\ln (e^{\tan x}+\tan x)}{\tan x}}}{x^3}]$
$=l[\frac{e^{\frac{\ln (1+2\sin x+\frac{{\sin }^{2}x}{2}+\frac{{\sin }^{3}x}{6}+\frac{{\sin }^{4}x}{24})}{\sin x}}-e^{\frac{\ln (1+2\tan x+\frac{{\tan }^{2}x}{2}+\frac{{\tan }^{3}x}{6}+\frac{{\tan }^{4}x}{24})}{\tan x}}}{x^3}]$
$=l[\frac{e^{2-\frac{3}{2}\sin x+\frac{11}{6}{\sin }^{2}x-\frac{57}{24}{\sin }^{3}x}-e^{2-\frac{3}{2}\tan x+\frac{11}{6}{\tan }^{2}x-\frac{57}{24}{\tan }^{3}x}}{x^3}]$
$=e^{2}l[\frac{p(\sin x)-p(\tan x)}{x^3},p(x)=-\frac{3}{2}x+\frac{71}{24}{x}^2-\frac{91}{16}{x}^3]$
$=e^{2}l[\frac{-\cos x\cdot p^{\prime }(\sin x)+{\cos }^{3}x\cdot p^{\prime \prime \prime }(x)-[2{\sec }^{3}x\cdot p^{\prime }(\tan x)+{\sec }^{6}x\cdot p^{\prime \prime \prime }(\tan x)]}{6}]$
$=-e^2\cdot \frac{p^{\prime }(0)}{2}$
$=\frac{3}{4}{e}^2$

【BV1jk4y127Ta】
$l[(\frac{a^x+b^x+c^x}{3}{)}^{\frac{1}{x}}]$
$=l[\exp (\frac{\ln \frac{1}{3}(a^x+b^x+c^x)}{x})]$
$=l[\exp (\frac{\ln \frac{1}{3}(1+x\ln a+1+x\ln b+1+x\ln c+o(x))}{x})]$
$=l[\exp (\frac{\ln (1+\frac{1}{3}x\ln abc+o(x))}{x})]$
$=l[\exp (\frac{\frac{1}{3}x\ln abc+o(x)}{x})]$
$=\sqrt[3]{abc}$

【BV1kA411T7KH】
$\underset{x\to 0}{\lim }(\frac{x}{\ln (1+x)}{)}^{\frac{1}{x}}$
$=\underset{x\to 0}{\lim }{e}^{\frac{\ln x-\ln \ln (1+x)}{x}}$
$=\underset{x\to 0}{\lim }{e}^{\frac{1}{x}-\frac{1}{(1+x)\ln (1+x)}}$
$=\underset{x\to 0}{\lim }{e}^{\frac{(1+x)\ln (1+x)-x}{x(1+x)\ln (1+x)}}$
$=\underset{x\to 0}{\lim }{e}^{\frac{(1+x)(x-\frac{x^2}{2}+o(x^2))-x}{x^2+o(x^2)}}$
$=\underset{x\to 0}{\lim }{e}^{\frac{\frac{1}{2}{x}^2+o(x^2)}{x^2+o(x^2)}}$
$=e^{\frac{1}{2}}$

【BV1rU4y147nc】
$\underset{x\to 0}{\lim }\frac{e^{(1+x{)}^{\frac{1}{x}}}-(1+x{)}^{\frac{e}{x}}}{x^2}$
$=\underset{x\to 0}{\lim }\frac{e^{\exp (\frac{\ln (1+x)}{x})}-e^{\frac{e\ln (1+x)}{x}}}{x^2}$
$=\underset{x\to 0}{\lim }\frac{e^{\exp (1-\frac{x}{2}+\frac{x^2}{3}+o(x^2))}-e^{e(1-\frac{x}{2}+\frac{x^2}{3}+o(x^2))}}{x^2}$
$=\underset{x\to 0}{\lim }\frac{e^{e[1+(-\frac{x}{2}+\frac{x^2}{3})+\frac{1}{2}(-\frac{x}{2}+\frac{x^2}{3}{)}^2+o(x^2)]}-e^{e(1-\frac{x}{2}+\frac{x^2}{3}+o(x^2))}}{x^2}$
$=\underset{x\to 0}{\lim }{e}^{e(1-\frac{x}{2}+\frac{x^2}{3})}\cdot \frac{e^{\frac{x^2}{8}+o(x^2)}-e^{o(x^2)}}{x^2}$
$=\underset{x\to 0}{\lim }{e}^e\cdot \frac{1+\frac{e{x}^2}{8}+o(x^2)-(1+o(x^2))}{x^2}$
$=\frac{1}{8}{e}^{e+1}$

【BV1tE411j7cB】
$\underset{x\to 0}{\lim }(1+\frac{1}{x}{)}^x$
$=\underset{x\to 0}{\lim }{e}^{x\ln (1+\frac{1}{x})}$
$=\underset{x\to 0}{\lim }{e}^{\frac{\ln (1+\frac{1}{x})}{\frac{1}{x}}}$
$=\underset{x\to 0}{\lim }{e}^{\frac{-\frac{1}{x^2}\frac{1}{(1+\frac{1}{x})}}{-\frac{1}{x^2}}}$
$=\underset{x\to 0}{\lim }{e}^{\frac{x}{1+x}}$
$=1$

【BV1QK4y1E7】
$\underset{x\to 0}{\lim }\frac{f^{\prime }(x)-\frac{f(x)-f(0)}{x}}{x}$
$=\underset{x\to 0}{\lim }\frac{1}{x}([f^{\prime }(0)+x{f}^{\prime \prime }(0)+o(x)]-\frac{1}{x}[f(0)+x{f}^{\prime }(0)+\frac{x^2}{2}{f}^{\prime \prime }(0)+o(x^2)-f(0)])$
$=\underset{x\to 0}{\lim }\frac{1}{x}([f^{\prime }(0)+x{f}^{\prime \prime }(0)]-\frac{1}{x}[x{f}^{\prime }(0)+\frac{x^2}{2}{f}^{\prime \prime }(0)]+o(x))$
$=\underset{x\to 0}{\lim }\frac{1}{x}(\frac{x}{2}{f}^{\prime \prime }(0)+o(x))$
$=\frac{1}{2}{f}^{\prime \prime }(0)$

【BV1Si4y1T7dK】
$\underset{x\to 0}{\lim }\frac{\tan (\tan (\cdots \tan x){)}_n-\sin (\sin (\cdots \sin x){)}_n}{x^3}$
$=\underset{x\to 0}{\lim }\frac{\tan (\cdots \tan (x+\frac{1}{3}{x}^3+o(x^3)){)}_{n-1}-\sin (\cdots \sin (x-\frac{1}{6}{x}^3+o(x^3)){)}_{n-1}}{x^3}$
$=\underset{x\to 0}{\lim }\frac{\tan (\cdots \tan (x+\frac{2}{3}{x}^3+o(x^3)){)}_{n-2}-\sin (\cdots \sin (x-\frac{2}{6}{x}^3+o(x^3)){)}_{n-2}}{x^3}$
$=\cdots$
$=\underset{x\to 0}{\lim }\frac{(x+\frac{n}{3}{x}^3+o(x^3))-(x-\frac{n}{6}{x}^3+o(x^3))}{x^3}$
$=\frac{n}{2}$

【BV1T5411G7nW】
$\underset{x\to 0}{\lim }\frac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{x^2}$
$=\underset{x\to 0}{\lim }\frac{1-[1-\frac{x^2}{2}+o(x^2)][1-x^2+o(x^2)][1-\frac{3}{2}{x}^2+o(x^2)]}{x^2}$
$=\underset{x\to 0}{\lim }\frac{1-(1-\frac{x^2}{2})(1-x^2)(1-\frac{3}{2}{x}^2)+o(x^2)}{x^2}$
$=\underset{x\to 0}{\lim }\frac{1-(1-\frac{x^2}{2}-x^2-\frac{3}{2}{x}^2)+o(x^2)}{x^2}$
$=\frac{1}{2}+1+\frac{3}{2}$
$=3$

【BV1Ti4y1F7aM】
$\underset{x\to 0}{\lim }\frac{x^2-{\arctan }^{2}x}{x^4}$
$=\underset{x\to 0}{\lim }\frac{x^2-(x-\frac{x^3}{3}+o(x^3){)}^2}{x^4}$
$=\underset{x\to 0}{\lim }\frac{x^2-(x^2-\frac{2}{3}{x}^4+o(x^4))}{x^4}$
$=\frac{2}{3}$

【BV1Xr4y1A7Ba】
$\underset{x\to 0}{\lim }(\sin 2x+\cos x{)}^{\frac{1}{x}}$
$=\underset{x\to 0}{\lim }(2x+\frac{(2x{)}^3}{3!}+o(x^3)+1-\frac{x^2}{2!}+o(x^2){)}^{\frac{1}{x}}$
$=\underset{x\to 0}{\lim }(1+2x+o(x){)}^{\frac{1}{x}}$
$=\underset{x\to 0}{\lim }{e}^{\frac{1}{x}\ln (1+2x+o(x))}$
$=\underset{x\to 0}{\lim }{e}^{2+o(1)}$
$=e^2$

趋于无穷

【BV1jk4y127Ta】
$L^{+}[(\begin{array}{c}\underline{a^x+b^x+c^x}\\ 3\end{array}{)}^{\frac{1}{x}}]$
$=L^{+}[\max (a,b,c)(\begin{array}{c}\underline{(\frac{a}{\max (a,b,c)}{)}^x+(\frac{b}{\max (a,b,c)}{)}^x+(\frac{c}{\max (a,b,c)}{)}^x}\\ 3\end{array}{)}^{\frac{1}{x}}]$
$=L^{+}[\max (a,b,c)(\frac{1+0+0}{3}{)}^{\frac{1}{x}}]$
$=\max (a,b,c)$

【BV1kG4y1C75e】
$L^{+}[n(e-(1+\frac{1}{n}{)}^n)]$
$=l^{+}[\frac{1}{n}(e-(1+n{)}^{\frac{1}{n}})]$
$=-l^{+}[((1+n{)}^{\frac{1}{n}}{)}^{\prime }]$
$=-l^{+}[(e^{\frac{1}{n}\ln (1+n)}{)}^{\prime }]$
$=-l^{+}[(e^{1-\frac{n}{2}+o(n)}{)}^{\prime }]$
$=-l^{+}[(-\frac{1}{2}+o(1))(e^{1-\frac{n}{2}+o(n)})]$
$=\frac{e}{2}$

【BV1mu411D7ZD】
$L^{+}[\frac{(1+\frac{1}{x}{)}^{x^2}}{e^x}]$
$=L^{+}[e^{x^2\ln (1+\frac{1}{x})-x}]$
$=l^{+}[e^{\frac{1}{x^2}\ln (1+x)-\frac{1}{x}}]$
$=l^{+}[e^{\frac{1}{x^2}(x-\frac{x^2}{2}+\frac{x^3}{3}+o(x))-\frac{1}{x}}]$
$=l^{+}(e^{\frac{1}{x}-\frac{1}{2}+\frac{x}{3}+o(x)-\frac{1}{x}})$
$=e^{-\frac{1}{2}}$

【BV1KA411N7cb】
$L^{+}(\sin \sqrt{x^2+1}-\sin \sqrt{x^2-1})$
$=L^{+}(\cos \frac{\sqrt{x^2+1}+\sqrt{x^2-1}}{2}\sin \frac{\sqrt{x^2+1}-\sqrt{x^2-1}}{2})$
$=L^{+}(\cos \frac{2x}{2}\sin \frac{1}{\sqrt{x^2+1}+\sqrt{x^2-1}})$
$=L^{+}(\cos x\cdot 2\sin \frac{1}{2x})$
$=L^{+}(\frac{2}{2x}\cos x)$
$=0$

【BV1PT41127FX】
$L^{+}[(x^{\frac{1}{x}}-1{)}^{\frac{1}{\ln x}}]$
$=l^{+}[(x^{-x}-1{)}^{\frac{1}{-\ln x}}]$
$=\exp (-l^{+}[\frac{\ln (x^{-x}-1)}{\ln x}])$
$=\exp (l^{+}[\frac{x(1+\ln x)\cdot x^{-x}}{x^{-x}-1}])$
$=\exp (l^{+}[\frac{x(1+\ln x)}{1-x^x}])$
$=\exp (l^{+}[\frac{x\ln x}{1-e^{x\ln x}}])$
因为$l^{+}(x\ln x)=l^{+}(\frac{\ln x}{\frac{1}{x}})=l^{+}(\frac{\frac{1}{x}}{-\frac{1}{x^2}})=0^{-}$
所以原式$\stackrel{t=x\ln x}{====}\exp (\underset{t\to 0^{-}}{\lim }\frac{t}{1-e^t})=\exp (-\underset{t\to 0^{-}}{\lim }\frac{1}{e^t})=e^{-1}$

【BV1Qf4y14797】
$l(\frac{{\int }_0^{2x}|1-\frac{t}{x}|\sin t\mathrm{d}t}{1-\cos x})$
$=l(\frac{{\int }_0^x(1-\frac{t}{x})t\mathrm{d}t+{\int }_x^{2x}(\frac{t}{x}-1)t\mathrm{d}t}{\frac{1}{2}{x}^2+o(x^2)})$
$=l(\frac{[\frac{t^2}{2}-\frac{t^3}{3x}{]}_0^x+[\frac{t^3}{3x}-\frac{t^2}{2}{]}_x^{2x}}{\frac{1}{2}{x}^2+o(x^2)})$
$=l(\frac{\frac{x^2}{2}-\frac{x^2}{3}+\frac{8}{3}{x}^2-2x^2-\frac{x^2}{3}+\frac{x^2}{2}}{\frac{1}{2}{x}^2+o(x^2)})$
$=l(\frac{x^2}{\frac{1}{2}{x}^2+o(x^2)})$
$=2$

【BV1Xr4y1A7Ba】
$L[(\sin \frac{2}{x}+\cos \frac{1}{x}{)}^x]$
$=l[(\sin 2x+\cos x{)}^{\frac{1}{x}}]$
$=l[(1+2x+o(x){)}^{\frac{1}{2x+o(x)}\frac{2x+o(x)}{x}}]$
$=l(e^{\frac{2x+o(x)}{x}})$
$=e^2$