Why can't a rational function have both, a horizontal and an oblique asymptote?

Question

Why if a rational function has a horizontal asymptote, it excludes the possibility of this function to have an oblique azymptote, or viceversa?

Answer 1

A rational function $f(x)=\frac{p(x)}{q(x)}$, where $p,q$ are polynomials, has the line $y=mx+b$ as asymptote if and only if polynomial division of $p$ by $q$ leads to $mx+b+\frac{r(x)}{q(x)}$ with $\deg r<\deg q$. By the lower degree condition, the $\frac{r(x)}{q(x)}$ summand tends to zero as $x\to \infty$ (as well as $x\to-\infty$).

It is the uniqueness of the result of polynomial division that guarantees the uniqueness of linear asymptotes.

(Note that vertical asymptotes, i.e., poles, i.e., roots of $q(x)$, are a different matter).

Answer 2

Horizontal and oblique asymptotes describe the behavior of the function "at infinity". If there is a horizontal asymptote, then the behavior at infinity is that the function is getting ever closer to a certain constant. If there is an oblique asymptote, then the function is getting ever closer to a line which is going to infinity. A function can't go to a finite constant and infinity at the same time.

Also, you might have an image in your head of a graph with a horizontal and oblique asymptote. If you do, then I'll be a Hersey bar that the graph violates the vertical line test, and therefor is not the graph of a function.