Horizontal asymptote

Definition

A horizontal asymptote is a horizontal line that a graph approaches as the input values go to positive or negative infinity. It indicates the end behavior of a function's output.

5 Must Know Facts For Your Next Test

1. For a rational function $\frac{P(x)}{Q(x)}$, if the degree of $P(x)$ is less than the degree of $Q(x)$, the horizontal asymptote is $y = 0$.
2. If the degrees of $P(x)$ and $Q(x)$ are equal, the horizontal asymptote is given by the ratio of their leading coefficients.
3. When the degree of $P(x)$ is greater than the degree of $Q(x)$, there is no horizontal asymptote; instead, it may have an oblique asymptote.
4. Horizontal asymptotes can be crossed by the graph at finite points but serve as boundary lines for extreme values.
5. Horizontal asymptotes are crucial for understanding long-term trends in real-world applications modeled by rational functions.

Review Questions

• How do you determine if a rational function has a horizontal asymptote?
• What is the horizontal asymptote of a rational function where both numerator and denominator have equal degrees?
• Can a graph cross its horizontal asymptote? Explain.

Related terms

Vertical Asymptote: A vertical line $x = c$ where a function grows without bound as it approaches this line from either direction.

Oblique Asymptote: A slanted line that a graph approaches as the input values go to positive or negative infinity, occurring when the degree of the numerator is exactly one more than the degree of the denominator.

End Behavior: The behavior of a function's graph as it approaches positive or negative infinity; describes how outputs behave for very large or very small inputs.