Calculus I · Limits and Continuity · lesson

How to Disprove a Claimed Limit

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 $\varepsilon_0$ for which no possible $\delta$ works.

Worked example

A jump fails the formal definition

Let

f(x)= \begin{cases} 0,&x<0,\\ 1,&x\ge0. \end{cases}

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

Show worked solution

Choose

\varepsilon_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.

After the explanation

Use the section idea

Reading lens

How small must the input window be to force every allowed output into the requested tolerance band?

Mental model

Epsilon sets the demanded vertical accuracy; delta is the horizontal promise you choose so every permitted nearby input meets that demand.

Decision

Work backward from the desired output inequality, isolate an input-distance bound, then state a positive delta that is no larger than that bound.

Common trap

A proof must control every eligible input in the punctured window; checking examples or choosing delta after seeing the input is not enough.

Check yourself

Formal understanding means you can translate between bands, inequalities, and words, then verify the implication from delta to epsilon in forward order.

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