calculus - Proving limit at infinity of a rational function

I need to prove these statements:

Let $f(x)=\sum_{j=0}^{n}a_jx^j$ , $g(x)=\sum_{j=0}^{m}b_jx^j$.

• $$\deg(g)\gt \deg(f) \implies \lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}=0$$


• $$\deg(g)= \deg(f) \implies \lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}=\frac{a_n}{b_n}$$


• $$\deg(f)\gt \deg(g) \implies \lim_{x\rightarrow \infty}\frac{f(x)}{g(x)}=\pm \infty$$

Is there any proof that would help me in all three statements, so that my answer can be shorter? I'm pretty sure I know how to do it, but I am trying to think of a cleaver way to shorten my answer.

Thanks!