Calculus and Its Applications, Chapter 1, 1.2, Section 1.2, Problem 18

Determine the $\displaystyle \lim_{x \to 3} \frac{x^2 - 25}{x^2 - 5}$ by using the Theorem on Limits of Rational Functions.
When necessary, state that the limit does not exist.


$
\begin{equation}
\begin{aligned}
\lim_{x \to 3} \frac{x^2 - 25}{x^2 - 5} &= \frac{\displaystyle\lim_{x \to 3} x^2 - 25}{\displaystyle\lim_{x \to 3} x^2- 5}
&& \text{The limit of a quotient is the quotient of the limits}\\
\\
&= \frac{\displaystyle\lim_{x \to 3} x^2 - \lim_{x \to 3} 25}{\displaystyle\lim_{x \to 3} x^2 - \lim_{x \to 3} 5}
&& \text{The limit of a difference is the difference of the limits}\\
\\
&= \frac{\left( \displaystyle\lim_{x \to 3} x\right)^2 - 25}{\left( \displaystyle\lim_{x \to 3}x \right)^2 - 5}
&& \text{The limit of a power is the power of the limit and the limit of a constant is the constant}\\
\\
&= \frac{(3)^2 - 25}{(3)^2 - 5}
&& \text{Substitute 3}\\
\\
&= \frac{9 - 25}{9 - 5}\\
\\
&= \frac{-16}{4}\\
\\
&= -4
\end{aligned}
\end{equation}
$