Let $f$ and $g$ be polynomial functions given by
$$
f(x):=a_{n} x^{n}+\cdots+a_{1} x+a_{0} \text { and } g(x):=b_{m} x^{m}+\cdots+b_{1} x+b_{0},
$$
where $a_{n}, \ldots, a_{0}, b_{m}, \ldots, b_{0}$ are in $\mathbb{R}, a_{n} \neq 0$, and $b_{m} \neq 0 .$ Show that
$$
\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=\left\{\begin{array}{ll}
0 & \text { if } m>n \\
a_{m} / b_{m} & \text { if } m=n
\end{array}\right.
$$
In case $m<n$, show that
$\frac{f(x)}{g(x)} \rightarrow \infty$ as $x \rightarrow \infty$ if $\frac{a_{n}}{b_{m}}>0$, and $\frac{f(x)}{g(x)} \rightarrow-\infty$ as $x \rightarrow \infty$ if $\frac{a_{n}}{b_{m}}<0$