Q60 / 125 · BITSAT 2014 · Mathematics

$\lim _{x \rightarrow 0}(\operatorname{cosec} x)^{1 / \log x}$ is equal to :

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

Solution

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

Hence, the correct answer is option: 'C'.

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