Limit at infinity

Definition

A limit at infinity refers to the value that a function approaches as the input (usually denoted as $x$) grows without bound in either the positive or negative direction. It helps determine the end behavior of a function.

5 Must Know Facts For Your Next Test

1. The notation for a limit at infinity is $\lim_{{x \to \infty}} f(x)$ or $\lim_{{x \to -\infty}} f(x)$.
2. If $\lim_{{x \to \infty}} f(x) = L$, then the horizontal line $y = L$ is called a horizontal asymptote of the function.
3. Rational functions often have limits at infinity that can be found by comparing the degrees of the numerator and denominator polynomials.
4. Limits at infinity are crucial for understanding the end behavior of polynomial and rational functions.
5. The concept can be extended to infinite limits, where $f(x)$ grows without bound as $x$ approaches infinity.

Review Questions

• What does it mean if $\lim_{{x \to -\infty}} f(x) = 3$?
• How do you find the limit at infinity for a rational function?
• What is indicated by a horizontal asymptote in terms of limits at infinity?

Related terms

Horizontal Asymptote:

A horizontal line that a graph approaches as x goes to plus or minus infinity.

End Behavior:

The behavior of a function's graph as x approaches positive or negative infinity.

Infinite Limit: A situation where a function increases or decreases without bound as x approaches a certain value.