Example 9. Find $\lim _{x \rightarrow 3} \frac{x^{2}-9}{\sqrt{x+1}-2}$.
Solution. When $x=3$, both the numerator and the denominator of the fraction become zero. The denominator contains an irrational expression $\sqrt{x+1}$. To eliminate the irrationality in the denominator, we multiply the numerator and the denominator by $(\sqrt{x+1}+2)$. We get

$$
\begin{gathered}
\lim _{x \rightarrow 3} \frac{x^{2}-9}{\sqrt{x+1}-2}=\lim _{x \rightarrow 3} \frac{(x-3)(x+3)(\sqrt{x+1}+2)}{(\sqrt{x+1}-2)(\sqrt{x+1}+2)}= \\
=\lim _{x \rightarrow 3} \frac{(x-3)(x+3)(\sqrt{x+1}+2)}{(\sqrt{x+1})^{2}-2^{2}}=\lim _{x \rightarrow 3} \frac{(x-3)(x+3)(\sqrt{x+1}+2)}{(x+1)-4}= \\
=\lim _{x \rightarrow 3}(x+3)(\sqrt{x+1}+2)=(3+3)(\sqrt{3+1}+2)=24 .
\end{gathered}
$$

Remark. To resolve an indeterminate form of the type $\frac{0}{0}$, where the numerator or the denominator contains an irrationality, one should appropriately eliminate the irrationality.