一、用定积分计算下面的极限.
(1%2B%5Cfrac%7B2%7D%7Bn%7D)%20%5Ccdots%20(1%2B%5Cfrac%7Bn%7D%7Bn%7D)%20%5Cbigg)%5E%7B1%2Fn%7D#card=math&code=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cbigg%28%281%2B%5Cfrac%7B1%7D%7Bn%7D%29%281%2B%5Cfrac%7B2%7D%7Bn%7D%29%20%5Ccdots%20%281%2B%5Cfrac%7Bn%7D%7Bn%7D%29%20%5Cbigg%29%5E%7B1%2Fn%7D&id=nfnV3). [
]
#card=math&code=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cbigg%28%20%5Cfrac%7Bn%7D%7Bn%5E2%2B1%5E2%7D%2B%20%5Cfrac%7Bn%7D%7Bn%5E2%2B2%5E2%7D%5Ccdots%20%2B%20%5Cfrac%7Bn%7D%7Bn%5E2%2Bn%5E2%7D%5Cbigg%29&id=FdPhL). [
]
, 其中
. [
#card=math&code=1%2F%28p%2B1%29&id=YUAl5).]
%5E%7B-1%7D#card=math&code=%5Clim%7Bn%20%5Cto%20%5Cinfty%7D%20%5Csin%20%5Cfrac%7B%5Cpi%7D%7Bn%7D%20%5Csum%7Bk%3D1%7D%5En%20%5Cbigg%282%2B%5Ccos%20%5Cfrac%7Bk%20%5Cpi%7D%7Bn%7D%20%5Cbigg%29%5E%7B-1%7D&id=SJ7Hq) [
. 提示：找等价无穷小代换.]
二、用定积分中值定理求下面的极限：
%20%5C%2C%20dx#card=math&code=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cint_a%5Eb%20%5Cexp%28-nx%5E2%29%20%5C%2C%20dx&id=kOFTX), 其中
为常数. 
[全部为零]
三、设
在
上连续，而且
%20%5C%2C%20dx%3D%5Cint_a%5Eb%20x%20f(x)%20%5C%2C%20dx%3D0#card=math&code=%5Cint_a%5Eb%20f%28x%29%20%5C%2C%20dx%3D%5Cint_a%5Eb%20x%20f%28x%29%20%5C%2C%20dx%3D0&id=Szk0l). 求证：
在
#card=math&code=%28a%2C%20b%29&id=Pks1D)内至少有两个零点.
四、设
在
#card=math&code=%5B0%2C%20%2B%5Cinfty%29&id=o39G8)上连续，而且
%3Da#card=math&code=%5Clim%7Bx%20%5Cto%20%2B%5Cinfty%7D%20f%28x%29%3Da&id=KTiMP). 求证：%5C%2C%20dt%20%3D%20a#card=math&code=%5Clim%7Bx%20%5Cto%20%2B%5Cinfty%7D%20%5Cfrac%7B1%7D%7Bx%7D%20%5Cint_0%5Ex%20f%28t%29%5C%2C%20dt%20%3D%20a&id=YjPwX).
五、设
，求证：
%20%3E%20%5Cfrac%7B2(b-a)%7D%7Bb%2Ba%7D#card=math&code=%5Cln%5Cbig%28%5Cfrac%7Bb%7D%7Ba%7D%5Cbig%29%20%3E%20%5Cfrac%7B2%28b-a%29%7D%7Bb%2Ba%7D&id=LUT2Y). [提示：考虑函数
; 利用中值定理. ]
