Horizontal Asymptote

Definition

A horizontal asymptote is a horizontal line that a graph of a function approaches as the input value (x) approaches positive or negative infinity. It represents the limit of the function as it gets closer and closer to this line, without ever actually touching it.

5 Must Know Facts For Your Next Test

1. The horizontal asymptote of an exponential function can be determined by evaluating the limit of the function as the input (x) approaches positive or negative infinity.
2. For exponential functions of the form $f(x) = a^x$, the horizontal asymptote is the x-axis (y = 0) if $a < 1$, and positive or negative infinity if $a > 1$.
3. The horizontal asymptote of a rational function can be determined by evaluating the ratio of the degrees of the numerator and denominator polynomials.
4. Horizontal asymptotes are important in understanding the behavior of functions, particularly as they approach their limits.
5. Knowing the horizontal asymptote of a function can help in sketching its graph and understanding its long-term behavior.

Review Questions

• Explain how the value of the base 'a' in an exponential function $f(x) = a^x$ determines the horizontal asymptote of the function.
  • The value of the base 'a' in an exponential function $f(x) = a^x$ determines the horizontal asymptote of the function. If $a < 1$, the horizontal asymptote is the x-axis (y = 0), meaning the function approaches the x-axis as the input (x) approaches positive or negative infinity. If $a > 1$, the horizontal asymptote is positive or negative infinity, depending on the sign of 'a', meaning the function grows or decays exponentially without bound as the input (x) approaches positive or negative infinity.
• Describe the relationship between the horizontal asymptote and the limit of a function as the input (x) approaches positive or negative infinity.
  • The horizontal asymptote of a function is directly related to the limit of the function as the input (x) approaches positive or negative infinity. The horizontal asymptote represents the value that the function approaches as the input (x) gets closer and closer to positive or negative infinity, but never actually reaches. Evaluating the limit of the function as $x$ approaches positive or negative infinity can help determine the equation of the horizontal asymptote, which is a crucial step in understanding the behavior and graphing of the function.
• Analyze how the horizontal asymptote of a rational function is determined by the relative degrees of the numerator and denominator polynomials.
  • For a rational function $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials, the horizontal asymptote is determined by the relative degrees of the numerator and denominator polynomials. If the degree of $P(x)$ is less than the degree of $Q(x)$, then the horizontal asymptote is the x-axis (y = 0). If the degree of $P(x)$ is greater than the degree of $Q(x)$, then the horizontal asymptote does not exist. If the degrees of $P(x)$ and $Q(x)$ are equal, then the horizontal asymptote is the line $y = \frac{a}{b}$, where $a$ is the leading coefficient of $P(x)$ and $b$ is the leading coefficient of $Q(x)$.

Related terms

Asymptote:

An asymptote is a line that a curve approaches but never touches, either as the input (x) approaches positive or negative infinity.

Exponential Function:

An exponential function is a function in which the independent variable appears as an exponent, resulting in a graph that grows or decays exponentially.

Limit:

The limit of a function is the value that the function approaches as the input (x) gets closer and closer to a particular value, but may never actually reach.