Calculus I · Limits and Continuity · lesson

Epsilon-Delta Proofs for Quadratic Functions

Concept

Learning objectives

Use a preliminary restriction such as $|x-a|<1$ to bound a factor that depends on $x$ ; choose $\delta$ as the minimum of two requirements.

Nonlinear Functions: Controlling an Extra Factor

For a quadratic, factoring the output error creates an extra factor involving $x$ .

Worked example

Show worked solution

We need to make

|x^2-4|

small. Factor:

|x^2-4|=|x-2||x+2|.

The first factor is controlled by $\delta$ , but $|x+2|$ still depends on $x$ . First impose the simple restriction

|x-2|<1.

Then

1<x<3,

3<x+2<5,

and therefore

|x+2|<5.

Now

|x^2-4|=|x-2||x+2|<5|x-2|.

To make this less than $\varepsilon$ , require

|x-2|<\frac\varepsilon5.

We need both restrictions, so choose

\boxed{\delta=\min\left\{1,\frac\varepsilon5\right\}}.

Now write the forward proof. Let $\varepsilon>0$ , and choose the $\delta$ above. If

0<|x-2|<\delta,

then $|x-2|<1$ , so $|x+2|<5$ . Also, $|x-2|<\varepsilon/5$ . Hence

|x^2-4| =|x-2||x+2| <5\left(\frac\varepsilon5\right) =\varepsilon.

Therefore,

\boxed{\lim_{x\to2}x^2=4}.

Concept

The minimum notation means, "Use whichever restriction is smaller." If $\varepsilon/5$ is tiny, use it. If $\varepsilon/5$ is larger than $1$ , keep the local bound $1$ . Both jobs must be done at once.

Worked example

A shifted square

Prove

\lim_{x\to1}x^2=1.

Show worked solution

Factor the output error:

|x^2-1|=|x-1||x+1|.

If $|x-1|<1$ , then $0<x<2$ , so $|x+1|<3$ . Therefore,

|x^2-1|<3|x-1|.

Choose

\boxed{\delta=\min\left\{1,\frac\varepsilon3\right\}}.

Then

|x^2-1|<3\delta\le\varepsilon.

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.

Source & rights

Original instruction with traceable references.

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