Calculus I · Limits and Continuity · extension

Formal Definition of an Infinite Limit

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.

After the explanation

Use the section idea

Reading lens

Is the function growing without bound near a finite input, or settling into end behavior as the input grows?

Mental model

Vertical asymptotes describe local blow-up near an excluded finite input; end-behavior asymptotes describe the long-run trend as inputs grow in magnitude.

Decision

Near a denominator zero, build a sign chart for each side; at infinity, compare dominant powers or divide to expose the lasting term.

Common trap

Do not merge positive and negative infinity, and remember that square roots produce absolute values when factoring a large squared input.

Check yourself

You understand the section when you can predict signs and asymptotes before doing detailed algebra, then verify them with the expression.

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