Horizontal asymptote

Definition

A horizontal asymptote is a horizontal line that a graph of a function approaches as the input (x) either increases or decreases without bound. This line represents a value that the function will get infinitely close to but never actually reach.

5 Must Know Facts For Your Next Test

1. Horizontal asymptotes can be found by analyzing the degrees of the numerator and denominator in rational functions.
2. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y=0$.
3. If the degrees of the numerator and denominator are equal, the horizontal asymptote is $y=\frac{a}{b}$ where $a$ and $b$ are leading coefficients.
4. There can be at most one horizontal asymptote for any given rational function.
5. Horizontal asymptotes describe end behavior but do not necessarily indicate behavior close to the origin.

Review Questions

• How do you determine if a rational function has a horizontal asymptote?
• What happens to a rational function's graph as x approaches infinity if there is a horizontal asymptote?
• Can a rational function cross its horizontal asymptote? Explain why or why not.

Related terms

Vertical Asymptote: A vertical line $x=c$ where a function's value becomes unbounded, typically occurring where the denominator of a rational function equals zero.

End Behavior: The behavior of a graph as x approaches positive or negative infinity, often described using limits and asymptotes.

Oblique Asymptote: $y=mx+b$, an asymptote that occurs when the degree of the numerator is exactly one more than that of the denominator in a rational function.