Question

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

Answer 1

$\underset{m\to \infty }{lim}{\left(\text{cos}\frac{x}{m}\right)}^m,\left[1^{\infty }\text{form}\right]$
$=e^{\underset{m\to \infty }{lim}m(\cos \frac{x}{m}-1)}$
$=e^{\underset{m\to \infty }{lim}m(-2{\sin }^2\frac{x}{2m})}$
$=e^{\underset{m\to \infty }{lim}-2m(x/2m{)}^2\left(\frac{{\sin }^2\frac{x}{2m}}{(x/2m{)}^2}\right)}$
$=e^{\underset{m\to \infty }{lim}\frac{-x^2}{2m}}$
$=e^0=1$