Definition: An asymptote is a line that a graph gets closer and closer to without ever settling on it, as x or y grows without bound.
What kinds of asymptotes exist?
A vertical asymptote is a line x = a where the function values blow up toward positive or negative infinity. A horizontal asymptote is a line y = L that the graph approaches as x goes far to the right or left. A slant asymptote appears when the numerator has degree exactly one more than the denominator.
How do you find them?
- Vertical: set the reduced denominator equal to zero.
- Horizontal: compare degrees. Smaller top gives y = 0; equal degrees give the ratio of leading coefficients.
Worked example: f(x) = (6x2 + 1)/(3x2 – 12). The denominator is zero at x = 2 and x = -2, so those are vertical asymptotes. The degrees are equal and the leading coefficients are 6 and 3, so the horizontal asymptote is y = 2.
What do students get wrong?
Thinking a graph can never cross a horizontal asymptote. It can cross it in the middle; the asymptote only describes far-away behavior. Mathbot also reminds you that a canceled factor gives a hole, not an asymptote.
Where to learn it
- Rational Functions: Asymptotes, Holes, and Graphs – lesson
- Polynomial and Rational Functions: Zeros and Asymptotes – lesson