How to Find Asymptotes of a Rational Function

An asymptote is a line that a curve approaches but never reaches. Rational functions can have vertical asymptotes, horizontal asymptotes, or both — and knowing where they are is essential to understanding the behavior of the function.

Vertical Asymptotes

A vertical asymptote occurs where the denominator of a rational function equals zero and that factor does not cancel with the numerator. The function shoots toward \(\pm\infty\) as \(x\) approaches that value.

To find vertical asymptotes:

  1. Factor both numerator and denominator completely.
  2. Cancel any common factors.
  3. Set the remaining denominator equal to zero — those values are the vertical asymptotes.

Example 1

Find the vertical asymptote(s) of \(\displaystyle f(x) = \frac{x+3}{x-2}\).

The denominator \(x - 2 = 0\) when \(x = 2\). There are no common factors to cancel.

Vertical asymptote: \(x = 2\).

As \(x\) approaches 2 from the right, \(f(x) \to +\infty\). As \(x\) approaches from the left, \(f(x) \to -\infty\).

Graph of (x+3)/(x-2) showing vertical asymptote at x=2

Holes vs. Asymptotes

This is the most important distinction to get right. When a factor cancels from both numerator and denominator, the result is not an asymptote — it's a hole (also called a removable discontinuity). The function is simply undefined at that point, but the curve doesn't blow up there.

Example 2

Find the asymptotes of \(\displaystyle f(x) = \frac{x^2 - x - 6}{x^2 - 9}\).

Factor:

  • Numerator: \(x^2 - x - 6 = (x-3)(x+2)\)
  • Denominator: \(x^2 - 9 = (x-3)(x+3)\)

$$f(x) = \frac{(x-3)(x+2)}{(x-3)(x+3)}$$

The factor \((x-3)\) appears in both — cancel it:

$$f(x) = \frac{x+2}{x+3} \qquad (x \neq 3)$$

Now read off the results:

  • \(x = 3\): the factor cancelled → hole at \(x = 3\) (the function is undefined there, but no asymptote)
  • \(x = -3\): the remaining denominator equals zero → vertical asymptote at \(x = -3\)

There is no vertical asymptote at \(x = 3\). This is a common mistake — every zero of the original denominator must be checked against the factored form to determine whether it's a hole or an asymptote.

Horizontal Asymptotes

A horizontal asymptote describes what happens to \(f(x)\) as \(x \to \pm\infty\). It depends on the degrees of the numerator and denominator polynomials.

Let \(n\) = degree of numerator, \(d\) = degree of denominator:

Condition Horizontal Asymptote
\(n < d\) \(y = 0\)
\(n = d\) \(y = \dfrac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\)
\(n > d\) None (the function grows without bound)

Example 3

Find the horizontal asymptote of \(\displaystyle f(x) = \frac{3x^2 + 1}{x^2 - 5}\).

Both numerator and denominator have degree 2 (\(n = d\)). The leading coefficients are 3 and 1.

Horizontal asymptote: \(y = 3\).

Example 4

Find the horizontal asymptote of \(\displaystyle f(x) = \frac{4x + 1}{x^2 + 3}\).

Numerator degree = 1, denominator degree = 2. Since \(n < d\):

Horizontal asymptote: \(y = 0\).

Example 5

Find all asymptotes of \(\displaystyle f(x) = \frac{2x^2 - 8}{x^2 + x - 6}\).

Factor:

  • Numerator: \(2(x-2)(x+2)\)
  • Denominator: \((x-2)(x+3)\)

Cancel \((x-2)\):

$$\frac{2(x+2)}{x+3} \qquad (x \neq 2)$$

  • \(x = 2\): hole
  • \(x = -3\): vertical asymptote
  • Degrees both 1 after cancellation, leading coefficients 2 and 1: horizontal asymptote \(y = 2\)

Practice Problems

1. Find the vertical asymptote(s) of \(\dfrac{5}{x^2 - 4}\). Show answerFactor denominator: \((x-2)(x+2)\). Neither factor cancels with the numerator (constant 5). Vertical asymptotes: \(x = 2\) and \(x = -2\).

2. Find all asymptotes and holes of \(\dfrac{x^2 - 1}{x^2 + x - 2}\). Show answerFactor numerator: \((x-1)(x+1)\). Factor denominator: \((x-1)(x+2)\). Cancel \((x-1)\): hole at \(x = 1\), vertical asymptote at \(x = -2\). Degrees equal, leading coefficients both 1: horizontal asymptote \(y = 1\).

3. Find the horizontal asymptote of \(\dfrac{7x^3 - 2x}{3x^3 + x^2 + 5}\). Show answerSame degree (3) in numerator and denominator. Leading coefficients 7 and 3. Horizontal asymptote: \(y = \frac{7}{3}\).

4. A function has numerator degree 4 and denominator degree 2. Does it have a horizontal asymptote? Show answerNo. When \(n > d\), the function grows without bound and there is no horizontal asymptote. (There may be an oblique/slant asymptote, but that requires polynomial long division.)