Solutions

In the forward direction $(\Rightarrow)$, assume the limit is a real number. For contradiction, suppose that $m<n$. This means that

$$\lim_{x\to\infty}\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0} =\lim_{x\to\infty}\frac{a_nx^{n-m}+a_{n-1}x^{n-1-m}+\cdots+a_0x^{-m}}{b_m+b_{m-1}x^{-1}+\cdots+b_0x^{-m}} =\pm\infty,$$

depending on the sign of $a_n$. This contradicts the supposition that the limit exists.

In the reverse direction $(\Leftarrow)$, suppose $m\geq n$. We break it down into two subcases. If $m=n$, the limit equals $a_n/b_m$. On the other hand, if $m>n$, the limit equals $0$. Hence it exists in both cases.