Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 54

Suppose $r$ is a rational function, show that $\lim \limits_{x \to a} R(x) = R(a)$ for every number $a$ in the domain of $r$

Let $P(x)$ be a rational function



$
\begin{equation}
\begin{aligned}

R(x) =& \frac{P(x)}{Q(x)} &&; \text{ Suppose that $Q(x) \neq 0$}\\
\\
R(a) =& = \frac{P(a)}{Q(a)}\\
\\
\lim \limits_{x \to a} R(x) =& \frac{\lim \limits_{x \to a} P(x)}{\lim \limits_{x \to a} Q(x)} &&; \text{ Suppose that $\lim \limits_{x \to a} Q(x) \neq 0$
(applying the limit law)} \\
\\
\lim \limits_{x \to a} R(x) =& \frac{P(a)}{Q(a)}
\\
.: \lim \limits_{x \to a} R(x) = R(a)


\end{aligned}
\end{equation}
$