Why if a rational function has a horizontal asymptote, it

George Bray

George Bray

Answered question

2022-06-22

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

Answer & Explanation

drumette824ed

drumette824ed

Beginner2022-06-23Added 19 answers

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

Leah Pope

Beginner2022-06-24Added 7 answers

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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?