Limits at Infinity

Definition

Limits at infinity refer to the behavior of a function as the input approaches infinity or negative infinity. This concept helps in understanding how functions behave when they grow indefinitely in one direction and is crucial for analyzing horizontal asymptotes and end behavior.

5 Must Know Facts For Your Next Test

1. To evaluate a limit at infinity, you often simplify the function by dividing all terms by the highest power of x in the denominator.
2. If a rational function's degree in the numerator is less than that in the denominator, the limit at infinity is zero.
3. If the degrees are equal, the limit at infinity is determined by the ratio of their leading coefficients.
4. For functions that approach a constant value as x approaches infinity, that constant is considered a horizontal asymptote.
5. Understanding limits at infinity is essential for sketching graphs of rational functions and determining their long-term behavior.

Review Questions

• How do you determine the limit at infinity for a rational function?
  • To find the limit at infinity for a rational function, you first identify the degrees of the polynomial in both the numerator and denominator. If the degree of the numerator is less than that of the denominator, the limit will be zero. If they are equal, you take the ratio of their leading coefficients. If the degree of the numerator is greater than that of the denominator, the limit will approach infinity or negative infinity depending on the signs.
• What role do horizontal asymptotes play in understanding limits at infinity?
  • Horizontal asymptotes represent values that a function approaches as its input goes to positive or negative infinity. They provide crucial information about a function's end behavior and help visualize how it behaves over large intervals. If a function has a horizontal asymptote at y = L, it means that as x approaches infinity (or negative infinity), the function values get closer and closer to L.
• Evaluate how limits at infinity contribute to the overall understanding of a function's behavior across its domain.
  • Limits at infinity provide valuable insights into a function's long-term trends, helping to categorize its growth and decay patterns. By analyzing these limits, we can identify horizontal asymptotes and predict how a function behaves far away from its defined values. This understanding also aids in graphing functions accurately and recognizing potential real-world implications, like predicting population growth or decay in various contexts.

Related terms

Horizontal Asymptote: A horizontal line that a graph approaches as the input values go towards positive or negative infinity, indicating the limit of the function at infinity.

End Behavior: The description of how a function behaves as its input values become very large or very small, often summarized by limits at infinity.

One-Sided Limits: Limits that consider the behavior of a function as the input approaches a certain value from one side, either from the left or the right.