Q49 / 90 · KVPY 2015 (SB/SX) · Mathematics

The value of the limit $$ \lim _{x \rightarrow 0}\left(\frac{x}{\sin x}\right)^{6 / x^{2}} $$ is

  • Correct answer
    A
    e
  • B
    $\mathrm{e}^{-1}$
  • C
    $\mathrm{e}^{-1 / 6}$
  • D
    $\mathrm{e}^{6}$

Solution

$\lim _{x \rightarrow 0}\left(\frac{x}{\sin x}\right)^{6 / x^{2}}$ $e^{\lim _{x \rightarrow 0} \frac{6}{x^{2}}\left(\frac{x-\sin x}{\sin }\right)}$ $\lim _{x \rightarrow 0} \frac{6}{x^{2}}\left(\frac{x-\left(x-\frac{x^{3}}{3}\right)}{x}\right)$ $=\mathrm{e}$

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

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