A horizontal asymptote (HA) of a function is an imaginary horizontal line to which its graph appears to be very close but never touch. It is of the form y = some number. Here, “some number” is closely connected to the excluded values from the range. A rational function can have at most one horizontal asymptote. Easy way to find the horizontal asymptote of a rational function is using the degrees of the numerator (N) and denominators (D).
- If N < D, then there is a HA at y = 0.
- If N > D, then there is no HA.
- If N = D, then the HA is y = ratio of the leading coefficients.
Example: Find the horizontal asymptote (if any) of the function f(x) = (x2 + 5x + 6) / (x2 + x – 2).
Solution:
Here the degree of the numerator is, N = 2, and the degree of the denominator is, D = 2.
Since N = D, the HA is y = (leading coefficient of numerator) / (leading coefficient of denominator) = 1/1 = 1.
Thus, the HA is y = 1.
Slant (Oblique) Asymptotes of a Rational Function
A slant asymptote is also an imaginary oblique line to which a part of the graph appears to touch. A rational function has a slant asymptote only when the degree of the numerator (N) is exactly one greater than the degree of the denominator (D). Its equation is y = quotient that is obtained by dividing the numerator by denominator using the long division.
Example: Find the slant asymptote of the function f(x) = x2/(x+1).
Solution:
Here the degree of numerator is 2 and that of denominator = 1. So it has a slant asymptote.
Let us divide x2 by (x + 1) by long division (or we can use synthetic division as well).
Thus, the slant asymptote is y = x – 1.