Calculus I · Limits and Continuity · quiz

Infinite Limits Concept Quiz

Section 4 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 checkinfinite-q1

Evaluate $\lim_{x\to2^-}\frac1{x-2}$ .

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From the left, $x-2$ is a small negative number.

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Interactive checkvertical-q1

For $f(x)=1/(x-3)^2$ , enter the $x$ -value of the vertical asymptote.

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A vertical asymptote occurs where the denominator is zero and does not cancel.

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Interactive checkend-q1

Evaluate $\lim_{x\to\infty}\frac{2x^2+1}{5x^2-3}$ .

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The degrees match, so compare leading coefficients.

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Interactive checkradical-infinity-q1

Evaluate $\lim_{x\to-\infty}\frac{\sqrt{x^2+1}}x$ .

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Remember $\sqrt{x^2}=|x|=-x$ when $x<0$ .

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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

Is the function growing without bound near a finite input, or settling into end behavior as the input grows?

Mental model

Vertical asymptotes describe local blow-up near an excluded finite input; end-behavior asymptotes describe the long-run trend as inputs grow in magnitude.

Decision

Near a denominator zero, build a sign chart for each side; at infinity, compare dominant powers or divide to expose the lasting term.

Common trap

Do not merge positive and negative infinity, and remember that square roots produce absolute values when factoring a large squared input.

Check yourself

You understand the section when you can predict signs and asymptotes before doing detailed algebra, then verify them with the expression.

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.