$\lim _{n \rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\ldots+\sqrt{n-1}}{n \sqrt{n}}$ का मान ज्ञात कीजिए।

सीमा का मान ज्ञात कीजिए: $\mathop {\lim}\limits_{n \to \infty } \frac{\pi }{2n} \left( 1 + \cos \frac{\pi }{2n} + \cos \frac{2\pi }{2n} + \dots + \cos \frac{(n - 1)\pi }{2n} \right)$

सीमा का मान ज्ञात कीजिए: $\lim _{n \rightarrow \infty} \frac{3}{n}\left\{1+\sqrt{\frac{n}{n+3}}+\sqrt{\frac{n}{n+6}}+\sqrt{\frac{n}{n+9}}+\ldots+\sqrt{\frac{n}{n+3(n-1)}}\right\}$

$\mathop {\lim }\limits_{n \to \infty } \left[ {\frac{1}{n} + \frac{1}{{n + 1}} + \frac{1}{{n + 2}} + \dots + \frac{1}{{2n}}} \right] = $

$\lim _{n \rightarrow \infty} n^4\left[\frac{1}{n^5}+\frac{1}{\left(n^2+1\right)^{\frac{5}{2}}}+\frac{1}{\left(n^2+4\right)^{\frac{5}{2}}}+\frac{1}{\left(n^2+9\right)^{\frac{5}{2}}}+\ldots+\right]=$

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