Calculus I · Limits and Continuity · lesson

How to Disprove a Claimed Limit

Learn How to Disprove a Claimed Limit with plain-language explanations, guided examples, worked homework methods, interactive checks, and exam-style practice.

Course progress46 of 47 core pages46 of 47

Where this chapter fits

Chapter 6: Formal limits

Translate the intuitive neighborhood picture into epsilon-delta language, constructive proofs, graph windows, and counterexamples.

Reading lens: How small must the input window be to force every allowed output into the requested tolerance band? Keep that question in view while reading How to Disprove a Claimed Limit; the worked mathematics is evidence for the idea, not a substitute for it.

This page connects Reading Epsilon and Delta From a Graph to Limits and Continuity Unit Review. Read the explanation first, predict each example’s next move, and only then compare the written solution.

Proving a Limit Is Not a Claimed Number

To show

\lim_{x\to a}f(x)\ne L,

it is enough to find one positive tolerance $\eps_0$ for which no possible $\delta$ works.

Worked example

A jump fails the formal definition

Let

f(x)= \left\{\matrix{ 0,\quad x<0,\quad 1,\quad x\ge0. }\right.

Show that $\lim_{x\to0}f(x)$ does not equal $1/2$ .

Show worked solution

Choose

\eps_0=\frac14.

The band around $1/2$ is

\left(\frac14,\frac34\right).

But every left-side input near zero has output $0$ , which lies outside this band, and every right-side input has output $1$ , also outside the band.

No matter how small $\delta>0$ is chosen, the point $x=-\delta/2$ satisfies

0<|x|<\delta,

but

\left|f(x)-\frac12\right| =\left|0-\frac12\right| =\frac12 >\frac14.

Therefore the claimed limit cannot be $1/2$ . In fact, the left and right limits disagree, so no two-sided limit exists.

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 6: Formal limits

Translate the intuitive neighborhood picture into epsilon-delta language, constructive proofs, graph windows, and counterexamples.

Proving a Limit Is Not a Claimed Number

To show

\lim_{x\to a}f(x)\ne L,

it is enough to find one positive tolerance

\eps_0

for which no possible

\delta

works.

Worked example

A jump fails the formal definition

Let

f(x)= \left\{\matrix{ 0,\quad x<0,\quad 1,\quad x\ge0. }\right.

Show that $\lim_{x\to0}f(x)$ does not equal $1/2$ .

Show worked solution

Choose

\eps_0=\frac14.

The band around $1/2$ is

\left(\frac14,\frac34\right).

But every left-side input near zero has output $0$ , which lies outside this band, and every right-side input has output $1$ , also outside the band.

No matter how small $\delta>0$ is chosen, the point $x=-\delta/2$ satisfies

0<|x|<\delta,

but

\left|f(x)-\frac12\right| =\left|0-\frac12\right| =\frac12 >\frac14.

Therefore the claimed limit cannot be $1/2$ . In fact, the left and right limits disagree, so no two-sided limit exists.

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.

Chapter 6: Formal limits

Proving a Limit Is Not a Claimed Number

Related resources

Original instruction with traceable references.

Chapter 6: Formal limits

Proving a Limit Is Not a Claimed Number

Related resources

Original instruction with traceable references.