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