Find \(\lim\limits_{x\rightarrow -2}\dfrac{x^{2}+5x+6}{x^{2}-4}\).

Solution Since \(-2\) is not in the domain of the rational function, (3) cannot be used. But this does not mean that the limit does not exist! Factoring the numerator and the denominator, we find \[ \frac{x^{2}+5x+6}{x^{2}-4}=\frac{(x+2)(x+3)}{(x+2)(x-2)} \]

Since \(x\neq -2\), and we are interested in the limit as \(x\) approaches \(-2\), the factor \(x+2\) can be divided out. Then \[ \begin{eqnarray*} \lim_{x\rightarrow -2}\frac{x^{2}+5x+6}{x^{2}-4}\underset{\underset{\color{#0066A7}{\hbox{Factor}}}{\color{#0066A7}{\uparrow}}}{=}\lim_{x\rightarrow -2}\frac{(x+2)(x+3)}{(x+2)(x-2)} \underset{\underset{\underset{\color{#0066A7}{\hbox{Divide out ( x+2)}}}{\color{#0066A7}{\hbox{x ≠ -2}}}}{\color{#0066A7}{\uparrow}}} {=}\lim_{x\rightarrow -2}\frac{x+3}{x-2} \underset{\underset{\underset{\color{#0066A7}{\hbox{Rational Function}}}{\color{#0066A7}{\hbox{Use the Limit of a}}}}{\color{#0066A7}{\uparrow}}} {=}\dfrac{-2+3}{-2-2}=-\dfrac{1}{4} \end{eqnarray*} \]