Question

Find $\underset{x\to 0}{lim}{\left(\cos x\right)}^{\frac{1}{x^2}}$

Answer 1

$L={lim}_{x\to 0}{\left(\cos x\right)}^{\frac{1}{x^2}}$.

$\log L=\frac{1}{x^2}\log \cos x$

$\log L=\frac{\log \cos x}{x^2}$

Using $L-Hospital$ rule,

$\log L=\frac{-\frac{\sin x}{\cos x}}{2x}=\frac{-\tan x}{2x}$

Again $L-Hospital$ rule,

$\log L=\frac{-{\sec }^{2}x}{2}=\frac{-1}{2}$

$L=e^{\frac{1}{2}}⟹\frac{1}{\sqrt{e}}$