Explanation

(c)
Let $y = \lim_{x \rightarrow 0}(cosec\ x)^{1/\ \log\ x}$
$\Rightarrow \ \ \ \ \ \ \log{y =}\lim_{x \rightarrow 0}\frac{\log{cosec\ x}}{\log x}$
$= \lim_{x \rightarrow 0}\ \ \ \frac{- \cot x}{1/x}$ [by L’Hospital’s rule]
$= - \lim_{x \rightarrow 0}\frac{x}{\tan x} = - 1$
$\Rightarrow \ \ \ \ \log{y = - 1}$
$\Rightarrow \ \ \ \ \ \ \ \ y = \frac{1}{e}$