Question

The value of $\underset{x\to 0}{lim}(co{s}^{2}x{)}^{\frac{1}{si{n}^{2}x}}$ is:

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 $k=\frac{1}{si{n}^{2}x}(co{s}^{2}x-1)=\underset{x\to 0}{lim}\frac{-si{n}^{2}x}{si{n}^{2}x}=-1$

So,

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