A) e
B) \[1/e\]
C) \[\pi /4\]
D) \[4/\pi \]
Correct Answer: B
Solution :
Let \[P=\underset{x\to \infty }{\mathop{\lim }}\,\,{{\left( \frac{n\,\,!}{{{n}^{n}}} \right)}^{1/n}}\]\[=\underset{n\to \infty }{\mathop{\lim }}\,\,{{\left( \frac{1}{n}\,.\,\frac{2}{n}\,.\,\frac{3}{n}\,.\,\frac{4}{n}\,..........\frac{n}{n} \right)}^{1/n}}\] \[\therefore \,\,\,\log \,\,P=\frac{1}{n}\,\underset{n\to \infty }{\mathop{\lim }}\,\,\left( \log \frac{1}{n}+\log \frac{2}{n}+......+\log \frac{n}{n} \right)\] \[\log \,\,P=\underset{n\to \infty }{\mathop{\lim }}\,\,\sum\limits_{r=1}^{n}{{}}\frac{1}{n}\log \frac{r}{n}\] \[\log \,\,P=\int_{0}^{1}{{}}\log x\,dx=(x\,\log x-x)_{0}^{1}=(-1)\] Þ \[P=\frac{1}{e}\] .
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