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# 5.3: Limits to Infinity and Infinite Limits

• • Dan Sloughter
• Professor (Mathematics) at Furman University
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## 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

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