Chapter 9.2, Problem 38E

For Problems 37 and 38, let

$f(x)=\frac{a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{\dots }+a_{1}x+a_0}{b_m{x}^m+b_{m-1}{x}^{m-1}+\mathrm{\dots }+b_{1}x+b_0}$

be a rational function.

(a) If m > n, show that $\underset{x\to \infty }{\lim }f\left(x\right)=0$ and hence that

y = 0 is a horizontal asymptote.

(b) If m < n, find $\underset{x\to \infty }{\lim }f\left(x\right).$

What does this say about horizontal asymptotes?