$\lim _{n \rightarrow \infty}\left[\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\ldots+\frac{n}{n^2+n^2}\right]=$

$\mathop {Limit}\limits_{n \to \infty } \frac{1}{n} \left[ 1 + \sqrt {\frac{n}{n + 1}} + \sqrt {\frac{n}{n + 2}} + \sqrt {\frac{n}{n + 3}} + \dots + \sqrt {\frac{n}{n + 3(n - 1)}} \right]$ has the value equal to

If $\lim _{n \rightarrow \infty} \frac{1}{n} \log \left(\frac{(2 n)!}{n^n \cdot n!}\right)=\int_1^2 f(x) d x$,then $f(x)=$

For each positive integer $n$,let $y_n = \frac{1}{n} ((n+1)(n+2) \dots (n+n))^{\frac{1}{n}}$. For $x \in \mathbb{R}$,let $[x]$ be the greatest integer less than or equal to $x$. If $\lim_{n \rightarrow \infty} y_n = L$,then the value of $[L]$ is:

$\mathop {\lim }\limits_{n \to \infty } \,\left( {\frac{n}{{{n^2} + {1^2}}} + \frac{n}{{{n^2} + {2^2}}} + \frac{n}{{{n^2} + {3^2}}} + ... + \frac{n}{{{n^2} + {{(2n)}^2}}}} \right)$ is equal to

$\lim _{n}$ ${\rightarrow \infty} \frac{1}{n}\left[\sin \frac{\pi}{4}+\sin \frac{\pi}{12}\left(3+\frac{1}{n}\right)+\sin \frac{\pi}{12}\left(3+\frac{2}{n}\right)+\ldots+\sin \frac{\pi}{3}\right]=$