$\mathop {\lim }\limits_{n \to \infty } \,\left( {\frac{n}{{{n^2} + {1^2}}} + \frac{n}{{{n^2} + {2^2}}} + \frac{n}{{{n^2} + {3^2}}} + ... + \frac{n}{{{n^2} + {{(2n)}^2}}}} \right)$ का मान ज्ञात कीजिए।

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

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

$\mathop {\lim }\limits_{n \to \infty } {\left[ {\frac{{n!}}{{{n^n}}}} \right]^{1/n}}$ का मान ज्ञात कीजिए।

$\mathop {Lim}\limits_{n \to \infty } \frac{\pi }{{6n}}\left[ {{{\sec }^2}\left( {\frac{\pi }{{6n}}} \right) + {{\sec }^2}\left( {2 \cdot \frac{\pi }{{6n}}} \right) + \dots + {{\sec }^2}\left( {(n - 1)\frac{\pi }{{6n}}} \right) + \frac{4}{3}} \right]$ का मान किसके बराबर है?

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