Limit at infinity

Definition

The limit at infinity refers to the behavior of a function as the input approaches infinity or negative infinity. It is used to analyze how a function behaves as the values of its variable grow larger and larger in either direction. Understanding limits at infinity helps in determining horizontal asymptotes and the long-term trends of functions.

5 Must Know Facts For Your Next Test

1. When evaluating limits at infinity, if the degree of the numerator is less than the degree of the denominator, the limit approaches 0.
2. If the degree of the numerator is greater than the degree of the denominator, the limit at infinity is either positive or negative infinity, depending on the leading coefficients.
3. For rational functions where degrees are equal, the limit at infinity is determined by dividing the leading coefficients of the numerator and denominator.
4. Limits at infinity can also be applied to exponential and logarithmic functions to understand their growth rates compared to polynomial functions.
5. Identifying limits at infinity helps in sketching graphs by showing how functions behave as they extend towards positive or negative infinity.

Review Questions

• How can you determine whether a rational function approaches a finite limit or infinity as you evaluate its limit at infinity?
  • To determine if a rational function approaches a finite limit or infinity at infinity, compare the degrees of the numerator and denominator. If the degree of the numerator is less than that of the denominator, the limit will approach 0. If it is greater, then the limit will be either positive or negative infinity based on the leading coefficients. If both degrees are equal, then you find the limit by dividing their leading coefficients.
• Explain how horizontal asymptotes are related to limits at infinity and provide an example.
  • Horizontal asymptotes represent the values that a function approaches as its input becomes infinitely large or infinitely small. When evaluating limits at infinity, if a function has a horizontal asymptote at $y = c$, it means that as $x$ approaches positive or negative infinity, $f(x)$ approaches $c$. For example, for the function $f(x) = \frac{2x^2 + 3}{5x^2 - 1}$, both degrees are equal, so as $x$ approaches infinity, the limit is $\frac{2}{5}$, indicating a horizontal asymptote at $y = \frac{2}{5}$.
• Analyze how understanding limits at infinity can impact your ability to graph various functions.
  • Understanding limits at infinity is crucial for accurately graphing functions because it informs us about their end behavior. For instance, knowing whether a function tends towards zero, approaches a specific number, or increases indefinitely helps in sketching accurate graphs. This understanding allows you to identify horizontal asymptotes, which give important context about how the function behaves far from the origin. This information is especially useful when comparing growth rates between polynomial and exponential functions, helping to clarify their long-term behavior on a graph.

Related terms

Horizontal Asymptote: A horizontal line that a graph approaches as the input values approach infinity, indicating the end behavior of the function.

Infinite Limits: A scenario where the value of a function increases or decreases without bound as the input approaches a certain value.

Polynomial Functions: Functions that are expressed in the form of a polynomial equation, which can be analyzed for their end behavior using limits at infinity.