Example 2. Find $\lim _{x \rightarrow 3}\left(\frac{1}{x-3}-\frac{6}{x^{2}-9}\right)$.
Solution. Here we also have an indeterminacy of the form $(\infty-\infty)$ as $x \rightarrow 3$. We perform the subtraction of fractions and cancel the factor $x-3 \neq 0$; passing to the limit, we find

$$
\begin{gathered}
\lim _{x \rightarrow 3}\left(\frac{1}{x-3}-\frac{6}{x^{2}-9}\right)=\lim _{x \rightarrow 3}\left(\frac{x+3-6}{x^{2}-9}\right)=\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-9}= \\
=\lim _{x \rightarrow 3} \frac{1}{x+3}=\frac{1}{9}
\end{gathered}
$$