Math
Asymptote Calculator
Find vertical, horizontal, and slant asymptotes of a rational function P(x) / Q(x). Enter coefficients up to degree 4 and see holes, domain, and degrees.
R(x) = P(x) / Q(x)
Enter coefficients of the numerator P(x) and denominator Q(x), highest power first. Both polynomials are limited to degree 4.
Degree 2
x^2 - 4
Degree 1
x - 2
Asymptote summary
- Vertical asymptotes
- None
- Horizontal asymptote
- None
- Oblique asymptote
- y = x + 2
- Holes
- x = 2
Numerator degree
2
Denominator degree
1
Domain restrictions
x != 2
Frequently Asked Questions about the Asymptote Calculator
What is an asymptote of a rational function?
An asymptote is a line that the graph of R(x) = P(x) / Q(x) approaches but never touches as x grows large or nears a forbidden value. Rational functions can have vertical asymptotes (from values of x where Q(x) is 0 but P(x) is not), one horizontal asymptote (based on the degrees of P and Q), and at most one oblique or slant asymptote when the numerator's degree is exactly one higher than the denominator's.
How do I find vertical asymptotes from coefficients?
Solve Q(x) = 0 for real x, then drop any root that also makes P(x) = 0 (those are holes, not asymptotes). For example, with P(x) = 1 and Q(x) = x - 3, the only root of Q is x = 3, and P(3) = 1, so x = 3 is a vertical asymptote. With P(x) = x^2 - 4 and Q(x) = x - 2, the root x = 2 is shared, so there is a hole at x = 2 and no vertical asymptote there.
When does a rational function have a horizontal asymptote?
Compare the degrees of P and Q. If deg(P) is less than deg(Q), the horizontal asymptote is y = 0. If the degrees are equal, the asymptote is y equal to the ratio of leading coefficients, for example (2x + 5) / (3x - 1) has y = 2 / 3. If deg(P) is greater than deg(Q), there is no horizontal asymptote, though there may be a slant one when the gap is exactly 1.
What is a slant or oblique asymptote?
A slant asymptote is a non-horizontal line y = mx + b that the curve approaches at infinity. It exists only when deg(P) = deg(Q) + 1. To find it, divide P(x) by Q(x) with polynomial long division; the quotient is the slant line and the remainder dies off as x grows. For x^2 / (x - 1), the quotient is x + 1, so the slant asymptote is y = x + 1.
What is the difference between a hole and a vertical asymptote?
Both come from values where Q(x) = 0, but a hole appears when P(x) also equals 0 at that same point, so the factor cancels and the function is just undefined at one point instead of blowing up. A vertical asymptote occurs when the factor only appears in Q, leaving the function unbounded near that x. The calculator reports holes and verticals separately so you can sketch the graph correctly.