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 $y=\underset{x\to 0}{\lim }{\left(cosecx\right)}^{1/logx}$
Taking log on both sides, we get
$logy=\underset{x\to 0}{\lim }\frac{logcosecx}{logx}$ [$\frac{\infty }{\infty }$ form]
$=\underset{x\to 0}{\lim }\frac{-cotx}{1/x}$ (By L' Hopital rule)
$=-\underset{x\to 0}{\lim }\frac{x}{tanx}\left(\because cotx=\frac{1}{tanx}\right)$
$\Rightarrow logy=-1$
$\Rightarrow y=e^{-1}=\frac{1}{e}$
Hence, required limit $=\frac{1}{e}$