\(L = \lim\limits_{x \to 0} (\cos x)^{\frac 1{x^2}}\)
\(\log L = \frac 1{x^2} \log \cos x\)
\(\log L = \frac {\log \cos x}{x^2}\)
Using L−Hospital rule,
\(\log L = \frac{- \frac {\sin x}{\cos x}}{2x} = \frac {-\tan x}{2x}\)
Again L−Hospital rule,
\(\log L = \frac{- \sec^2x}{2} = \frac {-1}{2}\)
\(L = e^\frac 12\Rightarrow \frac 1{\sqrt e}\)
