Calculus and Its Applications, Chapter 1, 1.2, Section 1.2, Problem 20

Determine the $\displaystyle \lim_{x \to 5} \frac{x^2 - 25}{x^2 - 5}$ 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 } \frac{x^2 - 25}{x^2 - 5} &= \lim_{x \to 5} \frac{(x+5)(x-5)}{x-5}
&& \text{Factor the numerator by applying the rule for difference of square}\\
\\
&= \lim_{x \to 5} (x + 5)
&& \text{Cancel out like terms}\\
\\
&= \lim_{x \to 5} x + \lim_{x \to 5} 5
&& \text{The limit of a sum is the sum of the limits}\\
\\
&= \lim_{x \to 5} x + 5
&& \text{The limit of a constant is the constant}\\
\\
&= 5 + 5
&& \text{Substitute } 5\\
\\
&= 10
\end{aligned}
\end{equation}
$