First slide

Introduction to limits

Question

The value of $lim_{x \to 0} {(\frac{\sin x}{x})}^{\frac{1}{x^{2}}},$ is

Moderate

A

$e^{- 1 / 3}$
B

$e^{1 / 3}$
C

$e^{- 1 / 6}$
D

$e^{1 / 6}$

Solution

We have,
$lim_{x \to 0} {(\frac{\sin x}{x})}^{\frac{1}{x^{2}}}$
$= lim_{x \to 0} {(x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots .)}^{1 / x^{2}}$
$= lim_{x \to 0} {(1 - \frac{x^{2}}{3!} + \frac{x^{4}}{5!} - \frac{x^{6}}{7!} + \dots)}^{\frac{1}{x^{2}}}$
$= e^{lim_{x \to 0} \frac{1}{x^{2}} (\frac{- x^{2}}{3!} + \frac{x^{4}}{5!} \frac{x^{6}}{7!} + \dots)} = e^{- 1 / 2!} = e^{- 1 / 6}$

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