Question
Evaluate $\lim \limits_{x \rightarrow 0} \frac{\sin a x}{b x}$

Answer

Given, $\lim _{x \rightarrow 0} \frac{\sin a x}{b x}$
$$Applying the limits in the given expression we get,$\lim _{x \rightarrow 0} \frac{\sin a x}{b x}=\frac{0}{0}$
Multiplying and dividing the given expression by a we get,
$\Rightarrow \lim _{x \rightarrow 0} \frac{\sin a x}{b x} \times \frac{a}{a}$
$\Rightarrow \lim _{x \rightarrow 0} \frac{\sin a x}{a x} \times \frac{a}{b}$
We know that: $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
$= \frac{a}{b} \lim _{a x \rightarrow 0} \frac{\sin a x}{a x}=\frac{a}{b} \times 1=\frac{a}{b}$

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