Horizontal, and Oblique Asymptotes
Main Concept

An asymptote is a line that the graph of a function approaches as either x or y approaches infinity. There are three types of asymptotes: vertical, horizontal and oblique.

Vertical Asymptotes


Vertical Asymptote

A vertical asymptote is a vertical line, , that has the property that either:
1.
2.
That is, as approaches from either the positive or negative side, the function approaches infinity.
Vertical asymptotes occur at the values where a rational function has a denominator of 0. The function is undefined at these points.





Horizontal Asymptotes


Horizontal Asymptote

A horizontal asymptote is a horizontal line, , that has the property that either:
1.
2.
Horizontal asymptotes occur when the numerator of a rational function has degree less than or equal to the degree of the denominator. If the denominator has degree , the horizontal asymptote can be calculated by dividing the coefficient of the th term of the numerator (it may be 0 if the numerator has a smaller degree) by the coefficient of the th term of the denominator.





Oblique Asymptotes


Oblique Asymptote

An oblique or slant asymptote is an asymptote along a line , where . Oblique asymptotes occur when the degree of the denominator of a rational function is one less than the degree of the numerator.







Use the sliders to choose the values , , and in the equation and see how they affect the horizontal and oblique asymptotes.

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