Question

$\lim\limits_{x\to0}\left(cosec \, x\right)^{1/log\, x}$ is equal to

• A. 0
• B. 1
• C. $\frac{1}{e}$
• D. None of these

Answer 1

Answer: (C)

Let $p=\lim _{x \rightarrow 0}(\operatorname{cosec} x)^{1 / \log x} \left(\infty^{0} \text { form }\right)$
$\Rightarrow \log p =\lim _{x \rightarrow 0} \frac{1}{\log x}(\log \operatorname{cosec} x) $
$ =\lim _{x \rightarrow 0} \frac{\log (\operatorname{cosec} x)}{\log x} \left(\frac{\infty}{\infty} \text { form }\right) $
$ \therefore \log p =\lim _{x \rightarrow 0}\left(\frac{-x \operatorname{cosec} x \cot x}{\operatorname{cosec} x}\right)$
(by L' Hospital's rule)
$=\lim _{x \rightarrow 0}\left(-\frac{x}{\tan x}\right)=-1$
$\therefore p=e^{-1}=\frac{1}{e}$