Calculus I · Limits and Continuity · quiz

Epsilon-Delta Concept Quiz

Section 6 Concept Quiz

Use this as a short retrieval check after finishing the section. On the website, each item should accept an answer before revealing the hint and worked explanation.

Interactive checkepsilon-linear-q1

For $f(x)=3x+1$ at $x=2$ , give a working choice of $\delta$ in terms of $\varepsilon$ .

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$|f(x)-7|=3|x-2|$ .

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Interactive checkepsilon-linear-q2

For $f(x)=5x$ at $x=1$ , give a working $\delta$ .

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$|5x-5|=5|x-1|$ .

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Interactive checkepsilon-quadratic-q1

For the standard proof that $x^2\to4$ as $x\to2$ , enter the usual safe choice of $\delta$ .

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First require $|x-2|<1$ , which makes $|x+2|<5$ .

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Interactive checkformal-reading-q1

In $|f(x)-L|<\varepsilon$ , does $\varepsilon$ control input distance or output distance?

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Look at which quantities are being compared.

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Summary

Quiz use on the website

This page should report completion by concept, not merely a total score. A wrong answer should link back to the exact lesson that teaches the missing method. The same questions may appear in randomized numerical variants, but the canonical explanation should remain stable.

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.

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.