# 5.3: Limits to Infinity and Infinite Limits

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## Definition

Let $$D \subset \mathbb{R}, f: D \rightarrow \mathbb{R},$$ and suppose $$a$$ is a limit point of $$D$$. We say that $$f$$ diverges to $$+\infty$$ as $$x$$ approaches $$a$$, denoted

$\lim _{x \rightarrow a} f(x)=+\infty ,$

if for every real number $$M$$ there exists a $$\delta>0$$ such that

$f(x)>M \text { whenever } x \neq a \text { and } x \in(a-\delta, a+\delta) \cap D.$

Similarly, we say that that $$f$$ diverges to $$-\infty$$ as $$x$$ approaches $$a,$$ denoted

$\lim _{x \rightarrow a} f(x)=-\infty ,$

if for every real number $$M$$ there exists a $$\delta>0$$ such that

$f(x)<M \text { whenever } x \neq a \text { and } x \in(a-\delta, a+\delta) \cap D.$

## Exercise $$\PageIndex{1}$$

Provide definitions for

a. $$\lim _{x \rightarrow a^{+}} f(x)=+\infty$$,

b. $$\lim _{x \rightarrow a^{-}} f(x)=+\infty$$,

c. $$\lim _{x \rightarrow a^{+}} f(x)=-\infty$$,

d. $$\lim _{x \rightarrow a^{-}} f(x)=-\infty$$.

Model your definitions on the preceding definitions.

## Exercise $$\PageIndex{2}$$

Show that $$\lim _{x \rightarrow 4^{+}} \frac{7}{4-x}=-\infty$$ and $$\lim _{x \rightarrow 4^{-}} \frac{7}{4-x}=+\infty$$.

## Definition

Suppose $$D \subset \mathbb{R}$$ does not have an upper bound, $$f: D \rightarrow \mathbb{R}$$, and $$L \in \mathbb{R} .$$ We say that the limit of $$f$$ as $$x$$ approaches $$+\infty$$ is $$L,$$ denoted

$\lim _{x \rightarrow+\infty} f(x)=L,$

if for every $$\epsilon>0$$ there exists a real number $$M$$ such that

$|f(x)-L|<\epsilon \text { whenever } x \in(M,+\infty) \cap D.$

## Definition

Suppose $$D \subset \mathbb{R}$$ does not have an lower bound, $$f: D \rightarrow \mathbb{R}$$, and $$L \in \mathbb{R} .$$ We say that the limit of $$f$$ as $$x$$ approaches $$-\infty$$ is $$L,$$ denoted

$\lim _{x \rightarrow-\infty} f(x)=L,$

if for every $$\epsilon>0$$ there exists a real number $$M$$ such that

$|f(x)-L|<\epsilon \text { whenever } x \in(-\infty, M) \cap D.$

## Exercise $$\PageIndex{3}$$

Verify that $$\lim _{x \rightarrow+\infty} \frac{x+1}{x+2}=1$$.

## Exercise $$\PageIndex{4}$$

Provide definitions for

a. $$\lim _{x \rightarrow+\infty} f(x)=+\infty$$,

b. $$\lim _{x \rightarrow+\infty} f(x)=-\infty$$,

c. $$\lim _{x \rightarrow-\infty} f(x)=+\infty$$,

d. $$\lim _{x \rightarrow-\infty} f(x)=-\infty$$.

Model your definitions on the preceding definitions.

## Exercise $$\PageIndex{5}$$

Suppose

$f(x)=a x^{3}+b x^{2}+c x+d,$

where $$a, b, c, d \in \mathbb{R}$$ and $$a>0 .$$ Show that

$\lim _{x \rightarrow+\infty} f(x)=+\infty \text { and } \lim _{x \rightarrow-\infty} f(x)=-\infty .$

This page titled 5.3: Limits to Infinity and Infinite Limits is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.