Horizontal asymptote

Definition

A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as x tends to positive or negative infinity. It indicates the behavior of the function at extreme values of x.

5 Must Know Facts For Your Next Test

1. Horizontal asymptotes are found by evaluating the limits of a function as x approaches positive or negative infinity.
2. If $\lim_{{x \to \infty}} f(x) = L$ or $\lim_{{x \to -\infty}} f(x) = L$, then $y = L$ is a horizontal asymptote.
3. A function can have different horizontal asymptotes as x approaches positive and negative infinity.
4. Rational functions, particularly those where the degrees of the numerator and denominator are equal, often have horizontal asymptotes determined by the leading coefficients.
5. Horizontal asymptotes do not necessarily indicate that the function will never cross them; they describe end behavior.

Review Questions

• How do you determine whether a function has a horizontal asymptote?
• Can a function have different horizontal asymptotes for $x \to \infty$ and $x \to -\infty$? Give an example.
• Why can a function cross its horizontal asymptote?

Related terms

Vertical Asymptote: A vertical line x = c where the function f(x) increases or decreases without bound as it approaches c.

Limit at Infinity: The value that a function approaches as the input variable grows without bound.

End Behavior: The behavior of a graph of a function as x approaches positive or negative infinity.