关于数学分析的计算题III(极限)

极限

$\bf计算：$$\lim \limits_{x \to \begin{array}{*{20}{c}}{ + \infty }\end{array}} \left( {\sin \sqrt {x + 1} - \sin \sqrt x } \right)$

1

$\bf计算：$$\lim \limits_{x \to \begin{array}{*{20}{c}}0\end{array}} \frac{{\cos \sqrt 2 x - {e^{ - {x^2}}} + \frac{4}{3}{x^4}}}{{{x^6}}}$

1

$\bf计算：$$\lim \limits_{n \to \infty } \sum\limits_{k = 1}^{{n^2}} {\frac{n}{{{n^2} + {k^2}}}} $

1

$\bf计算：$$\lim \limits_{n \to \infty } \left[ {\left( {1 + \frac{1}{n}} \right)\sin \frac{\pi }{{{n^2}}} + \left( {1 + \frac{2}{n}} \right)\sin \frac{{2\pi }}{{{n^2}}} +  \cdots  + \left( {1 + \frac{n}{n}} \right)\sin \frac{{n\pi }}{{{n^2}}}} \right]$

1

$\bf计算：$设$f(x)$在$[-1,1]$上连续且恒不为零，计算$\lim \limits_{x \to \begin{array}{*{20}{c}}0\end{array}} \frac{{\sqrt[3]{{1 + f\left( x \right)\sin x}} - 1}}{{{3^x} - 1}}$

1

$\bf计算：$设$f(x)$在$x=0$的邻域内有连续的一阶导数，且$f'\left( 0 \right) = 0,f''\left( 0 \right) = 1$，计算$\lim \limits_{x \to \begin{array}{*{20}{c}}0\end{array}} \frac{{f\left( x \right) - f\left( {\ln \left( {1 + x} \right)} \right)}}{{{x^3}}}$

1

$\bf计算：$$\lim \limits_{n \to \infty } n\left[ {{{\left( {1 + \frac{1}{n}} \right)}^n} - e} \right]$

1

$\bf计算：$设${a_1} > 0,{a_n} = \sin {a_{n - 1}}\left( {n \ge 2} \right)$，计算$\lim \limits_{n \to \infty } \sqrt {\frac{n}{3}} {a_n}$

1

$\bf计算：$$\lim \limits_{x \to \begin{array}{*{20}{c}}\infty\end{array}} \left[ {{{\left( {x - \frac{1}{2}} \right)}^2} - {x^4}{{\ln }^2}\left( {1 + \frac{1}{x}} \right)} \right]$ 

1

$\bf计算：$设$m \in Z$，计算$\lim \limits_{x \to \begin{array}{*{20}{c}}\infty \end{array}} {x^m}\int_0^{\frac{1}{x}} {\sin {t^2}dt} $

1

$\bf计算：$$\lim \limits_{t \to \begin{array}{*{20}{c}}0 \end{array}} \frac{{\int_0^t {\sin \left( {t{x^2}} \right)dx} }}{{{t^4}}}$

1

$\bf计算：$$\lim \limits_{n \to \infty } \left( {\frac{1}{{\sqrt {4{n^2} - {1^2}} }} + \frac{1}{{\sqrt {4{n^2} - {2^2}} }} +  \cdots  + \frac{1}{{\sqrt {4{n^2} - {n^2}} }}} \right)$

1

$\bf计算：$设${I_n} = \int_0^{\frac{\pi }{2}} {\frac{{{{\sin }^2}nt}}{{\sin t}}dt} \left( {n \in {N_ + }} \right)$，计算$\lim \limits_{n \to \infty } \frac{{{I_n}}}{{\ln n}}$

1

$\bf计算：$$\lim \limits_{n \to \infty } n\int_1^{1 + \frac{1}{n}} {\sqrt {1 + {x^n}} dx} $

1

$\bf计算：$$\lim \limits_{A \to  + \infty } \frac{1}{A}\int_0^A {\left| {\sin x} \right|dx} $

1

$\bf计算：$设$f\left( x \right) \in R\left[ {a,b} \right]$，计算$\lim \limits_{n \to \infty } \int_a^b {f\left( x \right)\left| {\sin nx} \right|dx} $

1

$\bf计算：$$\lim \limits_{n \to \infty } \left( {\frac{{\sin \pi /n}}{{n + 1}} + \frac{{\sin 2\pi /n}}{{n + \frac{1}{2}}} +  \cdots  + \frac{{\sin \pi }}{{n + \frac{1}{n}}}} \right)$

1