Horizontal Asymptote

Definition

A horizontal asymptote is a horizontal line that a graph of a function approaches as the input variable (usually x) approaches positive or negative infinity. It represents the limiting value that the function approaches, even though the function may never actually touch the asymptotic line.

5 Must Know Facts For Your Next Test

1. The existence of a horizontal asymptote is determined by the degree of the numerator and denominator of a rational function.
2. If the degree of the numerator is less than the degree of the denominator, the function will have a horizontal asymptote.
3. The value of the horizontal asymptote is equal to the ratio of the leading coefficients of the numerator and denominator polynomials.
4. Horizontal asymptotes are important in the analysis of the behavior of exponential and logarithmic functions as the input variable approaches positive or negative infinity.
5. Determining the horizontal asymptote of a function is a key step in sketching the graph of the function and understanding its long-term behavior.

Review Questions

• Explain how the degrees of the numerator and denominator of a rational function determine the existence and value of a horizontal asymptote.
  • The degrees of the numerator and denominator of a rational function $\frac{f(x)}{g(x)}$ determine the existence and value of the horizontal asymptote. If the degree of $f(x)$ is less than the degree of $g(x)$, then the function will have a horizontal asymptote at the value $\lim_{x\to\infty} \frac{f(x)}{g(x)} = \frac{a}{b}$, where $a$ is the leading coefficient of $f(x)$ and $b$ is the leading coefficient of $g(x)$. This is because as $x$ approaches positive or negative infinity, the lower-degree term in the numerator becomes negligible compared to the higher-degree term in the denominator.
• Describe the role of horizontal asymptotes in the analysis of exponential and logarithmic functions.
  • Horizontal asymptotes are important in understanding the behavior of exponential and logarithmic functions as the input variable approaches positive or negative infinity. For exponential functions of the form $f(x) = a^x$, the horizontal asymptote is the $x$-axis (y = 0) if $a > 1$, and the $x$-axis is also a horizontal asymptote for logarithmic functions of the form $f(x) = \log_a(x)$. Identifying the horizontal asymptote helps determine the long-term behavior of these functions and is crucial for sketching their graphs and interpreting their properties.
• Evaluate the significance of determining the horizontal asymptote of a function in the context of calculus and its applications.
  • Determining the horizontal asymptote of a function is a crucial step in the analysis and understanding of the function's behavior, which is essential in the study of calculus and its applications. Knowing the horizontal asymptote allows you to predict the long-term behavior of the function, which is important for sketching graphs, analyzing limits, and interpreting the function's real-world applications. Additionally, the horizontal asymptote provides insights into the relationship between the numerator and denominator of a rational function, which is necessary for understanding concepts like rational function optimization and the behavior of rational models in various disciplines, such as economics, physics, and engineering.

Related terms

Vertical Asymptote:

A vertical asymptote is a vertical line that a graph of a function approaches as the input variable (usually x) approaches a specific finite value.

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

Exponential Function: An exponential function is a function in which the independent variable appears as an exponent.