Limits at Infinity and Asymptotes

Formal Definition of a Limit Being Infinity

Let $f(x)$ be a function defined on an open interval $D$ that contains $c$ , except possibly at $x=c$ . Then we say that

$\lim _{x\to c}f(x)=\infty$

if, for every $\varepsilon$ , there exists a $\delta >0$ such that for all $x\in D$ with

$0<|x-c|<\delta$

we have

$f(x)>\varepsilon$ .

When this holds we write

$\lim _{x\to c}f(x)=\infty$

or

$f(x)\to \infty$ as $x\to c$

Similarly, we say that

$\lim _{x\to c}f(x)=-\infty$

if, for every $\varepsilon$ , there exists a $\delta >0$ such that for all $x\in D$ with

$0<|x-c|<\delta$

we have

$f(x)<\varepsilon$ .

When this holds we write

$\lim _{x\to c}f(x)=-\infty$

or

$f(x)\to -\infty$ as $x\to c$ .

Resources

• Limits at Infinity and Asymptotes PowerPoint file created by Dr. Sara Shirinkam, UTSA.

• Limits at Infinity and Asymptotes Worksheet 1

• Limits at Infinity and Asymptotes notes created by Instructor Beatty,UTSA.

• Limits at Infinity - Basic Idea and Shortcuts! by patrickJMT

• Calculus - Infinite Limits by patrickJMT

• How To Find The Limit At Infinity by the Organic Chemistry Tutor

• Limits at Infinity & Horizontal Asymptotes by the Organic Chemistry Tutor