Example 8. Find $\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}$.
Solution. By dividing the numerator and the denominator by $x$, based on formula (1.67) we obtain

$$
\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}=\lim _{x \rightarrow 0} \frac{\frac{\sin a x}{x}}{\frac{\sin b x}{x}}=\frac{\lim _{x \rightarrow 0} \frac{\sin a x}{x}}{\lim _{x \rightarrow 0} \frac{\sin b x}{x}}=\frac{a}{b}
$$

Therefore,

$$
\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}=\frac{a}{b}
$$