Question

$\underset{x\to 0}{\lim }{\left(cosecx\right)}^{1/logx}$ is equal to

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

Answer 1

Answer: (C)

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