# Horizontal, Vertical, & Oblique Asymptote?

So my precal assignment is asking me to find the vertical, horizontal, and oblique asymptotes.

I got to the following problem.

$\displaystyle R(X) = \frac{3x+5}{x-6}$

Would the horizontal asymptote be $y=3$ & vertical asymptote be $x=1$?

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Welcome to MSE! It helps to format using MathJax (see FAQ). I updated. Regards –  Amzoti Apr 15 '13 at 1:02

## 1 Answer

For a rational function of polynomials, which you have an example of (using just linear polynomials), the vertical asymptotes occur where the denominator is zero, provided the numerator is not also zero at the same value of x . So look again at your answer for that asymptote.

A rational function will have either a horizontal or an oblique asymptote, but not both. When the degrees of the numerator and denominator polynomials are the same, the asymptote of the function is "horizontal"; if the numerator's degree is larger, the asymptote is "oblique".* You can find the horizontal or oblique asymptote by polynomial division or by considering the ratio as |x| becomes very large (I am avoiding "limits at infinity" in this discussion since you said this is for a pre-calculus course). Your horizontal asymptote is correct.

*You have two linear functions, so the degrees are equal. If the numerator polynomial is higher in degree by 1 , the asymptote is a non-horizontal line and referred to as "oblique". If the numerator is higher in degree by more than 1, the asymptote is not a line, but a polynomial function. The non-horizontal asymptote functions can be found by polynomial division.

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would the vertical asymptote then be x = 11? Or am I just lost? –  Vernard Apr 15 '13 at 1:26
The denominator function is $x - 6$. For what value(s) of x is that equal to zero? If the numerator isn't also zero at any of those values, then you have located the vertical asymptotes. (The vertical asymptotes of a rational function occur where the function is not defined.) –  RecklessReckoner Apr 15 '13 at 1:57