Consider $f(x)=\frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials.

$($a$)$ If the degree of $p(x)$ is strictly smaller than the degree of $q(x)$ what is the limit of $f(x)$ as $x$ goes to infinity or negative infinity?

$($b$)$ If the degree of $p(x)$ is strictly larger than the degree of $q(x)$ what are the two values the limit of $f(x)$ can have as $x$ goes to infinity or negative infinity?

$($c$)$ How may you determine the limit as $x$ goes to infinity or negative infinity if the degree of $p(x)$ equals the degree of $q(x)?$