Example 7. Find $\lim _{x \rightarrow 0} \frac{\sin ^{2} \frac{x}{3}}{x^{2}}$.
Solution. Taking into account formula (1.67), based on the properties of limits, we obtain

$$
\lim _{x \rightarrow 0} \frac{\sin ^{2} \frac{x}{3}}{x^{2}}=\lim _{x \rightarrow 0}\left(\frac{\sin \frac{x}{3}}{x}\right)^{2}=\left[\lim _{x \rightarrow 0} \frac{\sin \frac{x}{3}}{x}\right]^{2}=\left(\frac{1}{3}\right)^{2}=\frac{1}{9}
$$