# 2.5: Limits at Infinity

- Page ID
- 48431

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Learning Objectives

- Calculate the limit of a function as \(x\) increases or decreases without bound.
- Recognize a horizontal asymptote on the graph of a function.

In this section, we define limits at infinity and show how these limits affect the graph of a function.

We begin by examining what it means for a function to have a finite limit at infinity. Then we study the idea of a function with an infinite limit at infinity.

## Key Concepts

- The limit of \(f(x)\) is \(L\) as \(x→∞\) (or as \(x→−∞)\) if the values \(f(x)\) become arbitrarily close to \(L\) as \(x\) becomes sufficiently large.
- The limit of \(f(x)\) is \(∞\) as \(x→∞\) if \(f(x)\) becomes arbitrarily large as \(x\) becomes sufficiently large. The limit of \(f(x)\) is \(−∞\) as \(x→∞\) if \(f(x)<0\) and \(|f(x)|\) becomes arbitrarily large as \(x\) becomes sufficiently large. We can define the limit of \(f(x)\) as \(x\) approaches \(−∞\) similarly.

## Glossary

**horizontal asymptote**- if \(\displaystyle \lim_{x→∞}f(x)=L\) or \(\displaystyle \lim_{x→−∞}f(x)=L\), then \(y=L\) is a horizontal asymptote of \(f\)

**infinite limit at infinity**- a function that becomes arbitrarily large as \(x\) becomes large

**limit at infinity**- a function that approaches a limit value \(L\) as \(x\) becomes large

## Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.