MCQ
$\mathop {\lim }\limits_{n \to \infty } {\left[ {\frac{{n!}}{{{n^n}}}} \right]^{1/n}}$ equals
- A$e$
- ✓$1/e$
- C$\pi /4$
- D$4/\pi $
$ = \mathop {\lim }\limits_{n \to \infty } \,{\left( {\frac{1}{n}\,.\,\frac{2}{n}\,.\,\frac{3}{n}\,.\,\frac{4}{n}\,..........\frac{n}{n}} \right)^{1/n}}$
$\therefore \,\,\,\log \,\,P = \frac{1}{n}\,\mathop {\lim }\limits_{n \to \infty } \,\left( {\log \frac{1}{n} + \log \frac{2}{n} + ...... + \log \frac{n}{n}} \right)$
$\log \,\,P = \mathop {\lim }\limits_{n \to \infty } \,\sum\limits_{r = 1}^n {} \frac{1}{n}\log \frac{r}{n}$
$\log \,\,P = \int_0^1 {} \log x\,dx = (x\,\log x - x)_0^1 = ( - 1)$
==> $P = \frac{1}{e}$ .
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Statement $1$ : $AB - BA$ is always an invertible matrix
Statement $2$ : $AB -BA$ is never an identity matrix