Mathematics - Limits Question with Solution | TestHub

Let $y={\lim }_{x\to 0}(\mathrm{cosec}x{)}^{1/\log x}$ Taking $\log$ on both sides, we get $\log y={\lim }_{x\to 0}\frac{\log \mathrm{cosec}x}{\log x}[\frac{\infty }{\infty }$ form $]$ $={\lim }_{x\to 0}\frac{-\cot x}{1/x}$ (ByL' Hospital rule) $=-{\lim }_{x\to 0}\frac{x}{\tan x}\left(\because \cot x=\frac{1}{\tan x}\right)$ $\Rightarrow \log y=-1$ $\Rightarrow y=e^{-1}=\frac{1}{e}$ Hence, required limit $=\frac{1}{e}$