$\lim _{n \rightarrow \infty} \left( \frac{1}{\sqrt{4n^2-1}} + \frac{1}{\sqrt{4n^2-4}} + \dots + \frac{1}{\sqrt{4n^2-n^2}} \right)$ ની કિંમત શોધો.

ધારો કે $S_n = \sum_{k=1}^n \frac{n}{n^2+kn+k^2}$ અને $T_n = \sum_{k=0}^{n-1} \frac{n}{n^2+kn+k^2}$ જ્યાં $n=1, 2, 3, \ldots$ છે. તો,

$\lim _{n \rightarrow \infty}\left(\frac{n^{2}}{\left(n^{2}+1\right)(n+1)}+\frac{n^{2}}{\left(n^{2}+4\right)(n+2)}+\frac{n^{2}}{\left(n^{2}+9\right)(n+3)}+\ldots+\frac{n^{2}}{\left(n^{2}+n^{2}\right)(n+n)}\right)$ ની કિંમત શોધો.

$\lim _{n \rightarrow \infty} \left[ \frac{n}{n^{2}+1^{2}} + \frac{n}{n^{2}+2^{2}} + \ldots + \frac{n}{n^{2}+n^{2}} \right]$ ની કિંમત શોધો.

$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{{n^3}}}\left[ {{1^2}\sin \frac{1}{n} + {2^2}\sin \frac{2}{n} + {3^2}\sin \frac{3}{n} + ....+{n^2}\sin \frac{n}{n}} \right]$ ની કિંમત શોધો.

$\lim _{n \rightarrow \infty} \frac{(2n(2n-1) \dots (n+1))^{1/n}}{n} = $