A rational function has at most 1 horizontal or oblique asymptote.
Compare the degree of the numerator polynomial () to the degree of the denominator polynomial ()
If then there is a horizontal asymptote at the line
If then there is a horizontal asymptote at the line where is the lead coefficient on the numerator polynomial and is the lead coefficient on the denominator polynomial.
If then there is an oblique asymptote defined by where is the lead coefficient on the numerator polynomial and is the lead coefficient on the denominator polynomial.
If then there is no horizontal or oblique asymptote, though there may be higher order curves to which the function is asymptotic.
Vertical Asymptotes
A rational function has a vertical asymptote at for every factor of the denominator polynomial that does not have a corresponding factor in the numerator. For example:
