一、计算(每小题10分,共70分)
1.求极限 $\displaystyle\lim\limits_{n\to +\infty}\prod_{k=1}^{n}\frac{4k-3}{4k}$.

2.计算$\displaystyle \displaystyle\lim\limits_{n\to +\infty}\sqrt[n]{n!} \ln{\left( 1+\frac{1}{n} \right) }$ .

3.对任意$\displaystyle A>0,f(x)$在$\displaystyle [0,A]$上可积,且$\displaystyle \lim\limits_{x\to +\infty}f(x)=1$,求$ \displaystyle \lim\limits_{T\to +\infty } \frac{1}{T}\int_{0}^{T}f(x)dx$.

4. 设$\displaystyle z=z(x,y)$由方程$\displaystyle e^{-xy}-2z+e^{z}=0$确定,求$\displaystyle \frac{\partial^{2}z}{\partial x\partial y}$.

5.求椭球面$\displaystyle \frac{x^{2}}{3}+\frac{y^{2}}{4}+\frac{z^{2}}{9}=1(x>0,y>0,z>0) $的切平面与三坐标平面所围成的几何体的最小体积.

6.设$f(t)$连续,$\displaystyle f(t)\sim t^{2}(t\to 0),F(t)=\iint_{x^{2}+y^{2}\le t}f(x^{2}+y^{2})dxdy \ \ (t\ge 0)$,求$\displaystyle F''(0+)$.

7.计算$\displaystyle \oint_{L}\left(y^{2}+z^{2} \right)dx +\left(z^{2}+x^{2} \right)dy+\left( x^{2}+y^{2}\right)dz$,其中$\displaystyle L$是曲面$\displaystyle x^{2}+y^{2}+z^{2}=4x$与$\displaystyle x^{2}+y^{2}=2x$的交线$\displaystyle z\ge 0$部分,曲线的方向规定从$z$轴正向看$L$为逆时针方向.

二、(10分)设$\displaystyle f(x)$在$[a,b]$上连续,且对任意$\displaystyle x\in [a,b],f(x)\not =0$,用定义法证明$\displaystyle \frac{1}{f(x)}$在$\displaystyle [a,b]$上一致连续.

三、(10分)  $\displaystyle f(x)$在$\displaystyle [0,1]$上可导,$\displaystyle f(1)=\int_{0}^{1}f(x)e^{1-x^{2}} dx$,证明:存在$\displaystyle a\in (0,1)$使得$\displaystyle f'(a)=2af(a)$.

四、(10分)令$\displaystyle f(x)=\frac{1}{x},a_{2n-1}=f(n),a_{2n}=\int_{n}^{n+1}f(x)dx$,讨论$\displaystyle \sum_{k=1}^{n}\left( -1\right) ^{n}a_{n}$的敛散性(收敛、绝对收敛和条件收敛).

五、(10分) 证明:$\displaystyle \int_{0}^{+\infty}\frac{\cos{x^2}}{1+x^{y}}dx$当:$\displaystyle y\in [0,+\infty]$时一致收敛.

六、(10分) $\displaystyle f(x)$在$\displaystyle [0,1]$上连续,证明:$\displaystyle \iint\limits_{S}f(x^{2}+y^{2})dS=2\pi \int_{-1}^{1}f(1-t^{2})dt$,其中$\displaystyle S$为球面$\displaystyle x^{2}+y^{2}+z^{2}=1$.

七、(15分) 设$ \displaystyle f(x) $和$\displaystyle g(x)$在$\displaystyle x_{0}$点附近恒正,判断下面命题$\displaystyle (A)$及其逆命题是否正确,若成立,给出证明；若不成立,举例说明.

$\displaystyle (A)$  : 当$\displaystyle x\to x_{0}$时,若$\displaystyle f(x)$和$\displaystyle g(x)$为等价无穷小量,则$\displaystyle \ln f(x)$和$\displaystyle \ln g(x)$为等价无穷大量.

八、(15分) 叙述Cauchy收敛定理,并用确界存在定理证明之.