Calculus I · Limits and Continuity · practice

Epsilon-Delta Practice Problems

Visual study stop

Read the picture before the symbols

Pause at each graph long enough to say what the input is doing, what the output is doing, and which feature supports the next mathematical decision.

Interactive epsilon band and delta window for f(x) = x squared over 2 plus 1 near (2, 3). — **Translate each proof into bands before manipulating symbols**
Begin with the requested vertical tolerance, work backward to a sufficient horizontal distance, and then verify forward that every input in the punctured delta window stays in the epsilon band.

Section 6 Exercises

Exercise 1

In your own words, explain the roles of $\varepsilon$ and $\delta$ .

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$\varepsilon$ is the allowed output error; $\delta$ is the input distance guaranteeing it.

Answer 1 from the source-traced unit appendix.

Exercise 2

Rewrite $|x-3|<0.2$ as an ordinary interval.

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$2.8<x<3.2$ .

Answer 2 from the source-traced unit appendix.

Exercise 3

Rewrite $|f(x)-5|<0.1$ as a vertical output interval.

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$4.9<f(x)<5.1$ .

Answer 3 from the source-traced unit appendix.

Exercise 4

For $f(x)=4x$ , find a $\delta$ in terms of $\varepsilon$ that proves $\lim_{x\to2}f(x)=8$ .

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$\delta=\varepsilon/4$ .

Answer 4 from the source-traced unit appendix.

Exercise 5

Prove $\lim_{x\to1}(5x-2)=3$ .

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$\delta=\varepsilon/5$ .

Answer 5 from the source-traced unit appendix.

Exercise 6

Prove $\lim_{x\to-2}(3x+4)=-2$ .

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$\delta=\varepsilon/3$ .

Answer 6 from the source-traced unit appendix.

Exercise 7

Prove $\lim_{x\to4}(x/2)=2$ .

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$\delta=2\varepsilon$ .

Answer 7 from the source-traced unit appendix.

Exercise 8

Prove $\lim_{x\to1}x^2=1$ using $\delta=\min\{1,\varepsilon/3\}$ .

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Use the proof in the section.

Answer 8 from the source-traced unit appendix.

Exercise 9

Prove $\lim_{x\to3}x^2=9$ by first bounding $|x+3|$ .

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One valid choice is $\delta=\min\{1,\varepsilon/7\}$ .

Answer 9 from the source-traced unit appendix.

Exercise 10

Find a suitable $\delta$ to prove $\lim_{x\to2}(x^2+x)=6$ .

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One valid choice is $\delta=\min\{1,\varepsilon/6\}$ .

Answer 10 from the source-traced unit appendix.

Exercise 11

Explain why a proof may choose a smaller $\delta$ than necessary.

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A smaller positive $\delta$ preserves the implication.

Answer 11 from the source-traced unit appendix.

Exercise 12

Explain why $\delta$ may depend on $\varepsilon$ , but not on the particular $x$ chosen later.

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The proof must guarantee the conclusion uniformly for every eligible $x$ in the window.

Answer 12 from the source-traced unit appendix.

Exercise 13

Use $\varepsilon_0=1/4$ to show the step function in the section does not approach $1/2$ at zero.

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Choose $\varepsilon_0=1/4$ ; points on either side remain at distance $1/2$ from the claimed limit.

Answer 13 from the source-traced unit appendix.

Exercise 14

Give a formal $M$ - $\delta$ proof that $1/x^2\to+\infty$ as $x\to0$ .

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Choose $\delta=1/\sqrt M$ .

Answer 14 from the source-traced unit appendix.

Exercise 15

Give a formal $N$ -style proof that $1/x\to0$ as $x\to\infty$ .

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Given $\varepsilon>0$ , choose $N>1/\varepsilon$ ; then $x>N$ implies $1/x<\varepsilon$ .

Answer 15 from the source-traced unit appendix.

Exercise 16

Identify the first incorrect step in the claim: "Choose $\delta=\varepsilon$ . Then $|x^2-4|<\varepsilon$ whenever $|x-2|<\delta$ ."

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The factor $|x+2|$ was not bounded, so $|x-2|<\varepsilon$ alone does not control the product.

Answer 16 from the source-traced unit appendix.

Exercise 17

Why is $0<|x-a|$ included in the formal finite-limit definition?

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A limit ignores the target point's value and studies punctured neighborhoods.

Answer 17 from the source-traced unit appendix.

Exercise 18

Explain how the graphical $\varepsilon$ -band and $\delta$ -window represent the quantified definition.

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The horizontal window must send the graph into the requested vertical band.

Answer 18 from the source-traced unit appendix.

Answers begin in the referenced section.

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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