Calculus I · Limits and Continuity · quiz

Trigonometric Limits Concept Quiz

Section 3 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 checksqueeze-q1

If $-x^2\le f(x)\le x^2$ , find $\lim_{x\to0}f(x)$ .

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Both outer functions approach the same value.

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Interactive checksine-q1

Evaluate $\lim_{x\to0}\frac{\sin(5x)}{x}$ .

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Make the denominator match the angle: multiply and divide by $5$ .

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Interactive checksine-ratio-q1

Evaluate $\lim_{x\to0}\frac{\sin(3x)}{\sin(7x)}$ .

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Build one $\sin u/u$ factor in the numerator and one in the denominator.

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Interactive checkcosine-q1

Evaluate $\lim_{x\to0}\frac{1-\cos x}{x^2}$ .

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Use the standard cosine limit or multiply by $1+\cos x$ .

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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

Can the expression be rewritten around a known small-angle limit, with every scaling factor accounted for?

Mental model

The fundamental sine limit is a reusable local shape: other trigonometric limits work when you expose that shape through identities and scaling.

Decision

Look for a bounded oscillation times a shrinking factor, or rewrite the expression into sine-over-angle factors whose arguments match their denominators.

Common trap

The sine function is not equal to its angle; their ratio merely approaches one near zero, and that statement requires radian measure.

Check yourself

Mastery means you can mark every scaling factor before simplifying and can explain where the Squeeze Theorem enters the argument.

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.