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:
- Factor both numerator and denominator completely.
- Cancel any common factors.
- 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, \(f(x) \to -\infty\).
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.)