Limit at Infinity

Definition

The limit at infinity refers to the behavior of a function as the input values approach infinity or negative infinity. This concept helps to determine how a function behaves as it grows larger and larger in the positive or negative direction, often indicating horizontal asymptotes. Understanding limits at infinity is crucial for analyzing the continuity and long-term behavior of functions.

5 Must Know Facts For Your Next Test

1. When evaluating limits at infinity for rational functions, if the degree of the numerator is less than the degree of the denominator, the limit is 0.
2. If the degrees of the numerator and denominator are equal, the limit at infinity is the ratio of their leading coefficients.
3. Limits at infinity can also be calculated for exponential functions, where exponential growth typically dominates polynomial growth.
4. Horizontal asymptotes can be identified by finding limits at infinity, indicating that a function approaches a specific value as the input becomes very large or very small.
5. When dealing with trigonometric functions, it's important to note that they may not have limits at infinity because they can oscillate indefinitely.

Review Questions

• How do you determine the limit at infinity for rational functions, and what does it signify about their behavior?
  • To find the limit at infinity for rational functions, compare the degrees of the numerator and denominator. If the degree of the numerator is less than that of the denominator, the limit approaches 0. If they are equal, the limit equals the ratio of their leading coefficients. This analysis helps to understand how rational functions behave as they grow larger, revealing key insights into their long-term trends.
• Discuss how horizontal asymptotes relate to limits at infinity and provide an example.
  • Horizontal asymptotes represent values that a function approaches as its input becomes infinitely large or small. To find horizontal asymptotes, one can evaluate limits at infinity. For example, for the function $$f(x) = \frac{2x^2 + 3}{4x^2 + 5}$$, as $$x$$ approaches infinity, both numerator and denominator have leading terms of degree 2. Thus, the limit is $$\frac{2}{4} = \frac{1}{2}$$ indicating a horizontal asymptote at $$y = \frac{1}{2}$$.
• Evaluate and analyze the significance of limits at infinity for exponential functions versus polynomial functions.
  • Exponential functions typically grow faster than polynomial functions as their input values increase. For instance, consider $$f(x) = e^x$$ compared to $$g(x) = x^2$$. As $$x$$ approaches infinity, $$e^x$$ tends toward infinity much faster than $$x^2$$. This highlights that while both types of functions may increase indefinitely, their rates of growth differ significantly; thus, understanding these limits can inform decisions about their long-term behavior in economic models or other applications.

Related terms

Asymptote: A line that a graph approaches but never touches, often used to describe the behavior of functions at infinity.

Continuous Function:

A function that has no breaks, jumps, or holes in its graph, meaning it can be drawn without lifting the pencil from the paper.

Infinite Series: A sum of infinitely many terms, which can converge to a finite value or diverge to infinity.