The graph of $f(x)$ has a horizontal asymptote of $y=a$ if
$$\lim_{x\to\infty}=a \text{ or }\lim_{x\to -\infty}=a$$
if $f\to a$ far away from the origin.
The graph of $f(x)$ has a vertical asymptote of $x=a$ if
$$\lim_{x\to a^+}=+\infty \text{ or }\lim_{x\to a^-}=+\infty \text{ or }
\lim_{x\to a^+}=-\infty \text{ or }\lim_{x\to a^-}=-\infty
$$
if the function "blows up" as $x\to a$.