First slide

Introduction to limits

Question

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

Moderate

A

$e^{- 1}$
B

$e$
C

1
D

none of these

Solution

We have,
$\begin{array}{l} lim_{x \to 0} {(\frac{\sin x}{x})}^{\frac{\sin x}{x - \sin x}} \\ = lim_{x \to 0} {(1 + \frac{\sin x - x}{x})}^{\frac{\sin x}{x - \sin x}} \\ = e^{lim_{x \to 0} \frac{\sin x - x}{x} \times \frac{\sin x}{x - \sin x}} = e^{- lim_{x \to 0} \frac{\sin x}{x}} = e^{- 1} \end{array}$

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