$\lim _{n \rightarrow \infty}\left[\frac{1}{n}+\frac{n}{(n+1)^{2}}+\frac{n}{(n+2)^{2}}+\ldots+\frac{n}{(2 n-1)^{2}}\right]$ is equal to ...... .

$\lim _{n \rightarrow \infty} n\left[\frac{1}{3 n^2+8 n+4}+\frac{1}{3 n^2+16 n+16}+\ldots+\frac{1}{15 n^2}\right]=$

The value of $\lim_{n \to \infty} \left[ \frac{1}{n}\sin \left( \frac{1}{n} \right)\left( \cos \left( \frac{1}{n} \right) \right)^2 + \frac{1}{n}\sin \left( \frac{2}{n} \right)\left( \cos \left( \frac{2}{n} \right) \right)^2 + \dots + \frac{1}{n}(\sin 1)(\cos 1)^2 \right]$ is

$\lim _{n \rightarrow \infty}\left[\frac{1}{n^2} \sec ^2 \frac{1}{n^2}+\frac{2}{n^2} \sec ^2 \frac{4}{n^2}+\ldots+\frac{1}{n} \sec ^2 1\right]=$

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

$\lim _{n \rightarrow \infty}\left(\frac{1}{1^2+n^2}+\frac{2}{2^2+n^2}+\frac{3}{3^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right)=$