Answer
$$
y= \frac{5+4 x}{x+3}
$$
We obtain that:
$y = 4$ is a horizontal asymptote.
$x=-3$ is a vertical asymptote.
The graph confirms our work.
Work Step by Step
$$
y= \frac{5+4 x}{x+3}
$$
we calculated that
$$
\lim _{x \rightarrow \pm \infty} \frac{5+4 x}{x+3}
$$
we divide the numerator and denominator by the highest
power of $x$ in the denominator, which is just $x$:
$$
\begin{aligned}
\lim _{x \rightarrow \pm \infty} \frac{5+4 x}{x+3}& =\lim _{x \rightarrow \pm \infty} \frac{(5+4 x) / x}{(x+3) / x}\\
&=\lim _{x \rightarrow \pm \infty} \frac{5 / x+4}{1+3 / x} \\
&=\frac{0+4}{1+0} \\
&=4
\end{aligned}
$$
so, $y = 4$ is a horizontal asymptote.
$$
y=f(x)= \frac{5+4 x}{x+3},
$$
so,
$$
\lim _{x \rightarrow -3^{+}}f(x)=-\infty
$$
since
$$
5+4 x \rightarrow-7 \text { and } x+3 \rightarrow 0^{+} \text {as } x \rightarrow-3^{+}
$$
and,
$$
\lim _{x \rightarrow -3^{-}}f(x)=\infty
$$
since
$$
5+4 x \rightarrow-7 \text { and } x+3 \rightarrow 0^{-} \text {as } x \rightarrow-3^{-}
$$
Thus $x=-3$ is a vertical asymptote. The graph confirms our work.
