Calculus and Its Applications, Chapter 1, 1.2, Section 1.2, Problem 12

Determine the $\displaystyle \lim_{x \to -2} (x^2 + 3)$ 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 -2} (x^2 + 3) &= \lim_{x \to -2} x^2 + \lim_{x \to -2} 3
&& \text{The limit of a sum is the sum of the limits}\\
\\
&= \left(\lim_{x \to -2} x \right)^2 + \lim_{x \to -2} 3
&& \text{The limit of a power is the power of the limit}\\
\\
&= \left(\lim_{x \to -2} x \right)^2 + 3
&& \text{The limit of a constant is the constant}\\
\\
&= (-2)^2 + 3
&& \text{Substitute } -2 \\
\\
&= 4 + 3
&& \text{Simplify}\\
\\
&= 7
\end{aligned}
\end{equation}
$