Calculus I · Limits and Continuity · extension

Formal Definition of an Infinite Limit

Go beyond the standard unit with Formal Definition of an Infinite Limit, including formal definitions, worked reasoning, and optional proof practice.

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Where this chapter fits

Chapter 7: Synthesis

Bring the unit together by choosing methods, explaining decisions, and correcting weak spots before an exam.

Reading lens: Can you diagnose the limit type and justify a method before beginning the algebra? Keep that question in view while reading Formal Definition of an Infinite Limit; the worked mathematics is evidence for the idea, not a substitute for it.

This page connects How to Disprove a Claimed Limit to Limits and Continuity Unit Review. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Formal Infinite Limits: Optional Extension

The statement

\lim_{x\to a}f(x)=+\infty

can be formalized as follows:

Definition

Formal Positive Infinite Limit

For every $M>0$ , there exists $\delta>0$ such that

0<|x-a|<\delta\implies f(x)>M.

Instead of asking for an output band around a finite $L$ , the challenger names an arbitrarily high threshold $M$ . The proof must force the function above it.

Guided walkthrough

$1/x^2\to+\infty$

Show formally that

\lim_{x\to0}\frac1{x^2}=+\infty.

Show worked solution

Let $M>0$ . We want

\frac1{x^2}>M.

This is equivalent to

x^2<\frac1M,

which is guaranteed by

|x|<\frac1{\sqrt M}.

Choose

\delta=\frac1{\sqrt M}.

Then $0<|x|<\delta$ implies

x^2<\delta^2=\frac1M,

\frac1{x^2}>M.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

The verified handoff declares original composition and requires owner provenance review. BetterGrades-original material remains separate from public-domain references; no source textbook PDF is published here.

Chapter 7: Synthesis

Bring the unit together by choosing methods, explaining decisions, and correcting weak spots before an exam.

Formal Infinite Limits: Optional Extension

The statement

\lim_{x\to a}f(x)=+\infty

can be formalized as follows:

Definition

Formal Positive Infinite Limit

For every $M>0$ , there exists $\delta>0$ such that

0<|x-a|<\delta\implies f(x)>M.

Instead of asking for an output band around a finite

L

, the challenger names an arbitrarily high threshold

M

. The proof must force the function above it.

Guided walkthrough

$1/x^2\to+\infty$

Show formally that

\lim_{x\to0}\frac1{x^2}=+\infty.

Show worked solution

Let $M>0$ . We want

\frac1{x^2}>M.

This is equivalent to

x^2<\frac1M,

which is guaranteed by

|x|<\frac1{\sqrt M}.

Choose

\delta=\frac1{\sqrt M}.

Then $0<|x|<\delta$ implies

x^2<\delta^2=\frac1M,

\frac1{x^2}>M.

Source & rights

Original instruction with traceable references.

The exposition is original. No Active Calculus exercise is reproduced verbatim. Public-domain examples were modernized and recomposed when used as inspiration.

Formal Definition of an Infinite Limit

Chapter 7: Synthesis

Formal Infinite Limits: Optional Extension

Formal Positive Infinite Limit

\(1/x^2\to+\infty\)

Original instruction with traceable references.

Formal Definition of an Infinite Limit

Chapter 7: Synthesis

Formal Infinite Limits: Optional Extension

Formal Positive Infinite Limit

\(1/x^2\to+\infty\)

Original instruction with traceable references.

Chapter 7: Synthesis

Formal Infinite Limits: Optional Extension

Related resources

Original instruction with traceable references.

Chapter 7: Synthesis

Formal Infinite Limits: Optional Extension

Related resources

Original instruction with traceable references.