Finding limits of some complicated expressions

Here in this post we are providing you the solutions of some complicated limits, these question are generally asked in CSIR,NBHM,GATE,TIFR,IIT-JAM, and other Compitative exams,.

What you need is to just click on the question whose answer you wants, then you will be redirected to its solution.

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\(\displaystyle 1).\;\; \lim_{n \to \infty}\frac{1}{\sqrt{n}}\left(\frac{1}{\sqrt{1}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+\cdots +\frac{1}{\sqrt{2n-1}+\sqrt{2n+1}}\right)\)

\(\displaystyle 2).\;\; \lim_{n \to \infty} n^{-{n}^2} \left[ \left (n+1 \right) \left( n+\frac{1}{2}\right) \left( n+\frac{1}{2^2}\right) \cdots \cdots \left(n+\frac{1}{2^{n-1}}\right) \right]^{n}\)

\(\displaystyle 3).\;\; \lim _{x \to 0} \frac{1}{x} \int_{x}^{2 x} e^{-t^{2}} d t \)

\(\displaystyle 4).\;\; \lim _{n \rightarrow \infty}\left(1-\frac{1}{n^{2}}\right)^{n} \)

\(\displaystyle 5).\;\; \lim _{n \rightarrow \infty} \frac{1}{\sqrt[n]{n !}} \)

\(\displaystyle 6).\;\; \lim _{n \rightarrow \infty} \frac{1}{n^{4}} \sum_{j=0}^{2 n-1} j^{3}\)

\(\displaystyle 7).\;\; \lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{n}{k^{2}+n^{2}}\)

\(\displaystyle 8).\;\; \lim _{n \rightarrow \infty} \Bigg\{\frac{2^{n+1}+3^{n+1}}{2^{n}+3^{n}}\Bigg\} \)

\(\displaystyle 9).\;\; \lim _{n \rightarrow \infty} \prod_{k=2}^{n}\left(1-\frac{1}{k^{2}}\right) \)

\(\displaystyle 10).\;\; S_{n}=\sum_{k=1}^{n} \displaystyle \frac{1}{\sqrt{4 n^{2}-k^{2}}}\)

\( \displaystyle 11).\;\; \lim _{n \rightarrow \infty}\left(\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2 n}\right) \)

\(\displaystyle 12).\;\; \lim _{n \rightarrow \infty}\left[\frac{1}{\sqrt{2 n^{2}-1}}+\cdots +\frac{1}{\sqrt{2 n^{2}-n}}\right] \)