Question

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

• A. $e$
• B. $e^{-1}$
• C. $e^{-1/6}$
• D. $e^6$

Answer 1

Answer: (A)

$e$

$\underset{x\to 0}{lim}{\left(\frac{x}{\sin x}\right)}^{6/x^2}$
$e^{\underset{x\to 0}{lim}\frac{6}{x^2}\left(\frac{x-\sin x}{\sin x}\right)}$
$e^{\underset{x\to 0}{lim}\frac{6}{x^2}\left(\frac{x-(x-\frac{x^3}{3!})}{x}\right)}$
$=e$.