关于数学分析的计算题III(极限)

极限

$f计算：$$lim limits_{x o egin{array}{*{20}{c}}{ + infty }end{array}} left( {sin sqrt {x + 1} - sin sqrt x } ight)$

1

$f计算：$$lim limits_{x o egin{array}{*{20}{c}}0end{array}} frac{{cos sqrt 2 x - {e^{ - {x^2}}} + frac{4}{3}{x^4}}}{{{x^6}}}$

1

$f计算：$$lim limits_{n o infty } sumlimits_{k = 1}^{{n^2}} {frac{n}{{{n^2} + {k^2}}}} $

1

$f计算：$$lim limits_{n o infty } left[ {left( {1 + frac{1}{n}} ight)sin frac{pi }{{{n^2}}} + left( {1 + frac{2}{n}} ight)sin frac{{2pi }}{{{n^2}}} +  cdots  + left( {1 + frac{n}{n}} ight)sin frac{{npi }}{{{n^2}}}} ight]$

1

$f计算：$设$f(x)$在$[-1,1]$上连续且恒不为零，计算$lim limits_{x o egin{array}{*{20}{c}}0end{array}} frac{{sqrt[3]{{1 + fleft( x ight)sin x}} - 1}}{{{3^x} - 1}}$

1

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

1

$f计算：$$lim limits_{n o infty } nleft[ {{{left( {1 + frac{1}{n}} ight)}^n} - e} ight]$

1

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

1

$f计算：$$lim limits_{x o egin{array}{*{20}{c}}inftyend{array}} left[ {{{left( {x - frac{1}{2}} ight)}^2} - {x^4}{{ln }^2}left( {1 + frac{1}{x}} ight)} ight]$ 

1

$f计算：$设$m in Z$，计算$lim limits_{x o egin{array}{*{20}{c}}infty end{array}} {x^m}int_0^{frac{1}{x}} {sin {t^2}dt} $

1

$f计算：$$lim limits_{t o egin{array}{*{20}{c}}0 end{array}} frac{{int_0^t {sin left( {t{x^2}} ight)dx} }}{{{t^4}}}$

1

$f计算：$$lim limits_{n o 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}} }}} ight)$

1

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

1

$f计算：$$lim limits_{n o infty } nint_1^{1 + frac{1}{n}} {sqrt {1 + {x^n}} dx} $

1

$f计算：$$lim limits_{A o  + infty } frac{1}{A}int_0^A {left| {sin x} ight|dx} $

1

$f计算：$设$fleft( x ight) in Rleft[ {a,b} ight]$，计算$lim limits_{n o infty } int_a^b {fleft( x ight)left| {sin nx} ight|dx} $

1

$f计算：$$lim limits_{n o infty } left( {frac{{sin pi /n}}{{n + 1}} + frac{{sin 2pi /n}}{{n + frac{1}{2}}} +  cdots  + frac{{sin pi }}{{n + frac{1}{n}}}} ight)$

1