Example 9. Find $\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x+9}-3}$.
Solution. When $x=0$, both the numerator and the denominator become zero. The denominator contains an irrationality. We will eliminate the irrationality and use formula (1.59)

$$
\begin{gathered}
\lim _{x \rightarrow 0} \frac{\sin x}{\sqrt{x+9}-3}=\lim _{x \rightarrow 0} \frac{\sin x(\sqrt{x+9}+3)}{(\sqrt{x+9}-3)(\sqrt{x+9}+3)}= \\
=\lim _{x \rightarrow 0} \frac{\sin x(\sqrt{x+9}+3)}{(x+9)-9}=\lim _{x \rightarrow 0} \frac{\sin x}{x} \cdot \lim _{x \rightarrow 0}(\sqrt{x+9}+3)=1(3+3)=6
\end{gathered}
$$