求下列极限
(1).$$lim_{n o infty}frac{1}{n}sum_{k=1}^{n}sin frac{kpi}{n}$$
(2).$$lim_{n o infty}left( frac{1}{n+1}+frac{2}{n+2}+cdots+frac{1}{n+n} ight)$$
(3).$$lim_{n o infty}left( frac{n}{n^{2}+1^{2}}+frac{n}{n^{2}+2^{2}}+cdots+frac{n}{n^{2}+n^{2}} ight)$$
(4).$$lim_{n o infty}frac{1^{p}+2^{p}+cdots+n^{p}}{n^{p+1}}$$
(5). $$lim_{n oinfty}frac{sqrt[n]{n!}}{n}$$
(6).$$lim_{n o infty}sum_{k=1}^{n}left(1+frac{k}{n} ight)sin frac{kpi}{n^{2}}$$
(7).$$lim_{n o infty}frac{1}{n}sqrt[n]{n(n+1)cdots (2n-1)}$$
(8).设$f(x)in C[0,1]$,$f(x)$处处大于$0$,求极限
$$lim_{n o infty}sqrt[n]{fleft(frac{1}{n} ight)fleft(frac{2}{n} ight)cdots fleft(frac{n-1}{n} ight)f(1)}$$