Question

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