Question

Evaluate: $\underset{x\to 0}{lim}[co{s}^{2}x{]}^{\frac{1}{si{n}^{2}x}}$

Answer 1

Given limit is of form $1^{\infty }$. Hence

$\underset{x\to 0}{lim}[co{s}^{2}x{]}^{\frac{1}{si{n}^{2}x}}=e^k$

Where

$k=(co{s}^{2}x-1)\frac{1}{si{n}^{2}x}=-1$

Hence,

$\underset{x\to 0}{lim}[co{s}^{2}x{]}^{\frac{1}{si{n}^{2}x}}=e^{-1}=\frac{1}{e}$