Horizontal Asymptotes

Definition

A horizontal asymptote is a horizontal line that a function's graph approaches as the input variable (typically x) approaches positive or negative infinity. It represents the limiting value that the function approaches but never actually reaches.

5 Must Know Facts For Your Next Test

1. Horizontal asymptotes are particularly important in the study of rational functions, where they can be used to determine the long-term behavior of the function.
2. The existence and location of horizontal asymptotes can be determined by examining the relative degrees of the numerator and denominator polynomials in a rational function.
3. If the degree of the numerator is less than the degree of the denominator, the function will have a horizontal asymptote at y = 0.
4. If the degree of the numerator is equal to the degree of the denominator, the function will have a horizontal asymptote at y = the ratio of the leading coefficients.
5. If the degree of the numerator is greater than the degree of the denominator, the function will not have a horizontal asymptote.

Review Questions

• Explain how the relative degrees of the numerator and denominator polynomials in a rational function determine the existence and location of the horizontal asymptote.
  • The relative degrees of the numerator and denominator polynomials in a rational function determine the existence and location of the horizontal asymptote. If the degree of the numerator is less than the degree of the denominator, the function will have a horizontal asymptote at y = 0. If the degree of the numerator is equal to the degree of the denominator, the function will have a horizontal asymptote at y = the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, the function will not have a horizontal asymptote.
• Describe how the concept of limits is related to the idea of horizontal asymptotes.
  • Horizontal asymptotes are closely related to the concept of limits. As the input variable (typically x) approaches positive or negative infinity, the function's output value approaches the horizontal asymptote. In other words, the horizontal asymptote represents the limiting value that the function approaches but never actually reaches. This connection between horizontal asymptotes and limits is crucial in understanding the long-term behavior of functions, particularly rational functions.
• Analyze the importance of horizontal asymptotes in the study of rational functions and their applications.
  • Horizontal asymptotes are particularly important in the study of rational functions because they can be used to determine the long-term behavior of the function. By examining the relative degrees of the numerator and denominator polynomials, one can identify the existence and location of the horizontal asymptote, which provides valuable information about the function's behavior as the input variable approaches positive or negative infinity. This understanding of the function's long-term behavior is crucial in various applications, such as modeling real-world phenomena, analyzing the performance of systems, and making informed decisions based on the function's properties.

Related terms

Asymptote: A line that a curve approaches but never touches, either as the input variable approaches positive or negative infinity.

Rational Function: A function that can be expressed as the ratio of two polynomial functions.

Limit: The value that a function approaches as the input variable approaches a particular value.