In the previous exercise, do $m$ and $n$ have to be integers, that is, must $f$ be a rational function?

Suppose the rational function $r(x)=q(x)+l(x)$ where $l(x)$ approaches 0 as $x$ approaches $+\infty$ or $-\infty,$ then $r(x) \rightarrow q(x)$ as $x$ approaches $+\infty$ or $-\infty,$ or we say $r(x)$ is asymptotic to $q(x) .$ For example, $$r(x)=\frac{2 x^{2}+6 x+8}{x+1}=2 x+4+\frac{4}{x+1}$$ as $x$ approaches $+\infty$ or $-\infty, 4 /(x+1) \rightarrow 0,$ so $r(x) \rightarrow 2 x+4 .$ Therefore $y=2 x+4$ is an asymptote for $r(x) .$ A sketch is given is Figure $34 .$ Note how the graph approaches the line $y=2 x+4$ as $x$ gets large. When $l(x)$ is linear it is called a slant or oblique asymptote.