Question

The value of $\underset{m\to \infty }{lim}{\left(\cos \frac{x}{m}\right)}^m$ is

• A. $1$
• B. $e$
• C. $e^{-1}$
• D. None of these

Answer 1

Answer: (A)

$1$

$\underset{m\to \infty }{lim}{\left(\cos \frac{x}{m}\right)}^m$
Let $y={\left(\cos \frac{x}{m}\right)}^m$
$\Rightarrow \log y=m\log \cos \frac{x}{m}$
$\Rightarrow y=e^{m\log \cos \frac{x}{m}}$
$\Rightarrow \underset{m\to \infty }{lim}y=e^{{lim}_{m\to \infty }m\log \cos \frac{x}{m}}$
$=e^{{lim}_{m\to \infty }\frac{\log \cos \frac{x}{m}}{1/m}}$
It is of the form $\frac{0}{0}$, so applying L-Hospital's rule
$\Rightarrow \underset{m\to \infty }{lim}y=e^{{lim}_{m\to \infty }x\tan \frac{x}{m}}$
$=e^0=1$
Hence, option 'A' is correct.