Calculus I · Limits and Continuity · lesson

Epsilon-Delta Proofs for Linear Functions

Concept

Learning objectives

Construct a $\delta$ directly from a requested $\varepsilon$ for a linear function.

Linear Functions: The Cleanest Proofs

Guided walkthrough

$f(x)=2x$

Prove

\lim_{x\to3}2x=6.

Show worked solution

We want the output error to be less than $\varepsilon$ :

|2x-6|<\varepsilon.

Factor:

|2x-6|=2|x-3|.

So it is enough to require

2|x-3|<\varepsilon.

Divide by $2$ :

|x-3|<\frac\varepsilon2.

Choose

\boxed{\delta=\frac\varepsilon2}.

Then if $0<|x-3|<\delta$ ,

|2x-6|=2|x-3|<2\delta=2\left(\frac\varepsilon2\right)=\varepsilon.

Therefore,

\lim_{x\to3}2x=6.

Interactive checkepsilon-flow-01

For $f(x)=2x$ at $x=3$ , choose $\delta$ in terms of $\varepsilon$ .

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Show hint

$|2x-6|=2|x-3|$ .

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This proof has two phases.

• Discovery phase. Start from the desired output inequality and work backward to guess a useful $\delta$ . • Proof phase. State the chosen $\delta$ , assume the input inequality, and work forward to the output inequality.

Worked example

A general linear function

Prove

\lim_{x\to2}(3x+1)=7.

Show worked solution

Discovery. We need

|(3x+1)-7|<\varepsilon.

Simplify:

|3x-6|=3|x-2|.

Thus, requiring

|x-2|<\frac\varepsilon3

is enough.

Proof. Let $\varepsilon>0$ . Choose

\delta=\frac\varepsilon3.

0<|x-2|<\delta,

then

|(3x+1)-7|=3|x-2|<3\delta=\varepsilon.

Therefore,

\boxed{\lim_{x\to2}(3x+1)=7}.

Summary

For $f(x)=mx+b$ , a standard choice is

\delta=\frac\varepsilon{|m|}

when $m\ne0$ . The output changes $|m|$ times as much as the input, so the input tolerance must be $|m|$ times smaller.

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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