Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 12

Determine the $\lim\limits_{x \rightarrow -4} \quad \displaystyle \frac{x^2+5x+4}{x^2+3x-4}$, if it exists



$
\begin{equation}
\begin{aligned}
\lim\limits_{x \rightarrow -4} \quad \displaystyle \frac{x^2+5x+4}{x^2+3x-4} &= \lim\limits_{x \rightarrow -4}
\frac{
\cancel{(x+4)}(x+1)
}
{
\cancel{(x+4)}(x-1)
}
&& \text{(Get the factors and cnacel out like terms)}\\
\lim\limits_{x \rightarrow -4} \quad \displaystyle \frac{x+1}{x-1} &= \frac{-4+1}{-4-1} && \text{(Substitute value of } x \text{ and simplify)}
\end{aligned}
\end{equation}\\
\boxed{\lim\limits_{x \rightarrow -4} \quad \displaystyle \frac{x^2+5x+4}{x^2+3x-4} = \frac{3}{5}}
$