First slide

Introduction to limits

Question

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

Moderate

A

$e^{\sin a}$
B

$e^{\tan a}$
C

$e^{\cot a}$
D

1

Solution

$\begin{array}{l} lim_{x \to a} {(\frac{\sin x}{\sin a})}^{\frac{1}{x - a}} \\ = lim_{x \to a} {\{1 + \frac{\sin x - \sin a}{\sin a}\}}^{\frac{1}{x - a}} \\ = e^{lim_{x \to a} \frac{\sin x - \sin a}{x - a} \times \frac{1}{\sin a}} = e^{\frac{\cos a}{\sin a}} = e^{\cot a} \end{array}$

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