极限

作者: Raow1 | 来源:发表于2021-02-07 01:54 被阅读0次

2017-1-2. 求极限 $\lim\limits_{x \to 0}\frac{1}{x^3} \int_0^x \sin{t^2} \mathrm dt$

$\begin{align*} \lim_{x \to 0}\frac{1}{x^3} \int_0^x \sin{t^2} \mathrm dt &= \lim_{x \to 0} \frac{\sin x^2}{3x^2} \\ &=\lim_{x \to 0} \frac{x^2}{3x^2} \\ &= \frac{1}{3} \end{align*}$

2017-2-2. 求极限 $\lim\limits_{x \to 0}\frac{1}{x} \int_0^x \frac{\sin{2t}}{t} \mathrm dt$

$\begin{align*} \lim_{x \to 0}\frac{1}{x} \int_0^x \frac{\sin{2t}}{t} \mathrm dt &= \lim_{x \to 0} \frac{\sin 2x}{x} \\ &=\lim_{x \to 0} \frac{2x}{x} \\ &= 2 \end{align*}$

2019-2-2. 求极限 $\lim\limits_{x \to 0} (\frac{\sin x}{x})^{\frac{1}{1-\cos x}}$

$\begin{align*} \lim_{x \to 0} (\frac{\sin x}{x})^{\frac{1}{1-\cos x}} &= \lim_{x \to 0} (1+\frac{\sin x -x}{x})^{\frac{x}{\sin x -x} \cdot \frac{\sin x -x}{x(1-\cos x)}} \\ &= e^{\lim\limits_{x \to 0} \frac{\sin x -x}{x(1- \cos x)}} \\ &= e^{\lim\limits_{x \to 0} \frac{-2x^3}{6x^3}} \\ &= e^{-\frac{1}{3}} \end{align*}$

2019-3-2. 求极限 $\lim\limits_{x \to +\infty} (\frac{\arctan x}{\arccos \frac{1}{x}})^{\frac{x}{1-x\sin \frac{1}{x}}}$

$\begin{align*} \lim_{x \to +\infty} (\frac{\arctan x}{\arccos \frac{1}{x}})^{\frac{x}{1-x\sin \frac{1}{x}}} &= e^{\lim\limits_{x \to +\infty} \frac{x}{1-x\sin \frac{1}{x}} \cdot \frac{\arctan x - \arccos \frac{1}{x}}{\arccos \frac{1}{x}} } \\ &\xlongequal{t=\frac{1}{x}} e^{\lim\limits_{t \to 0} \frac{\arctan{\frac{1}{t} - \arccos t}}{ (t-\sin t)\arccos t}} \\ &= e^{\lim\limits_{t \to 0} \frac{(\frac{\pi}{2}-t+\frac{t^3}{3} +o(t^3))-(\frac{\pi}{2}-t-\frac{t^3}{6} + o(t^3)) }{\frac{\pi}{2} \cdot \frac{x^3}{6}} } \\ &= e^{\frac{6}{\pi}} \end{align*}$

极限

2017-1-2. 求极限 $\lim\limits_{x \to 0}\frac{1}{x^3} \int_0^x \sin{t^2} \mathrm dt$

2017-2-2. 求极限 $\lim\limits_{x \to 0}\frac{1}{x} \int_0^x \frac{\sin{2t}}{t} \mathrm dt$

2019-2-2. 求极限 $\lim\limits_{x \to 0} (\frac{\sin x}{x})^{\frac{1}{1-\cos x}}$

2019-3-2. 求极限 $\lim\limits_{x \to +\infty} (\frac{\arctan x}{\arccos \frac{1}{x}})^{\frac{x}{1-x\sin \frac{1}{x}}}$

2019-4-2. 求极限 $\lim\limits_{x \to +\infty} |\ln (\tanh x)|^{arccot x}$

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