Calculus I · Limits and Continuity · exam

Limits and Continuity Practice Exam B

Answer key published

Finish first. Then check every answer.

The complete key is online, numbered to match this exam, and linked to the verified source appendix.

Practice Examination B

This exam is deliberately less patterned. It is meant to test whether you can identify methods rather than imitate section headings.

Suggested time: 90 minutes.

Exercise 1

Let

f(x)=\begin{cases} x^2+1,&x<1,\\ 5,&x=1,\\ 3x-1,&x>1. \end{cases}

Find both one-sided limits, the two-sided limit, $f(1)$ , and determine continuity at $1$ .

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Left $2$ , right $2$ , two-sided limit $2$ , function value $5$ ; not continuous, removable discontinuity.

Answer 1 from the source-traced unit appendix.

Exercise 2

$\displaystyle\lim_{x\to2}\frac{x^3-8}{x^2-4}$

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Cancel $x-2$ : $(x^2+2x+4)/(x+2)\to12/4=3$ .

Answer 2 from the source-traced unit appendix.

Exercise 3

$\displaystyle\lim_{x\to0}\frac{\sqrt{1+4x}-\sqrt{1-x}}x$

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Rationalization gives $5/[\sqrt{1+4x}+\sqrt{1-x}]\to5/2$ .

Answer 3 from the source-traced unit appendix.

Exercise 4

$\displaystyle\lim_{x\to0}\frac{1-\cos(3x)}{x^2}$

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$9\cdot(1/2)=9/2$ .

Answer 4 from the source-traced unit appendix.

Exercise 5

$\displaystyle\lim_{x\to0}\frac{\tan(2x)}{\sin(5x)}$

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$2/5$ .

Answer 5 from the source-traced unit appendix.

Exercise 6

$\displaystyle\lim_{x\to0}x\sin(1/x^2)$

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Squeeze gives $0$ .

Answer 6 from the source-traced unit appendix.

Exercise 7

Find every discontinuity of $\dfrac{(x-1)(x+2)}{(x-1)(x-3)^2}$ , classify it, and state all one-sided infinite limits.

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Hole at $1$ ; vertical asymptote $x=3$ . Both one-sided limits at $3$ are $+\infty$ .

Answer 7 from the source-traced unit appendix.

Exercise 8

$\displaystyle\lim_{x\to\infty}\frac{2x^3-x}{x^2+1}$

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$+\infty$ .

Answer 8 from the source-traced unit appendix.

Exercise 9

Find the polynomial asymptote of the function in the previous problem.

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Division gives polynomial asymptote $y=2x$ .

Answer 9 from the source-traced unit appendix.

Exercise 10

$\displaystyle\lim_{x\to-\infty}(\sqrt{x^2+10x}+x)$

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$-5$ .

Answer 10 from the source-traced unit appendix.

Exercise 11

Find $a,b$ so

g(x)=\begin{cases} ax+b,&x<1,\\ x^2+2,&1\le x<2,\\ 3x+a,&x\ge2 \end{cases}

is continuous at $1$ and $2$ .

Show answer

Continuity at $2$ gives $6=6+a$ , so $a=0$ ; continuity at $1$ then gives $a+b=3$ , so $b=3$ .

Answer 11 from the source-traced unit appendix.

Exercise 12

Show that $x^5-3x+1=0$ has a root in $(0,1)$ , then perform two bisection steps.

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The polynomial is continuous, $f(0)=1$ , $f(1)=-1$ . First midpoint $0.5$ is negative, so interval $(0,0.5)$ . Second midpoint $0.25$ is positive, so interval $(0.25,0.5)$ .

Answer 12 from the source-traced unit appendix.

Exercise 13

Prove $\lim_{x\to1}(4x-3)=1$ using the formal definition.

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Choose $\delta=\varepsilon/4$ ; then $|(4x-3)-1|=4|x-1|<4\delta=\varepsilon$ .

Answer 13 from the source-traced unit appendix.

Exercise 14

A student claims that because $f(-1)<0<f(1)$ , every function has a root in $(-1,1)$ . Give a counterexample and state the missing hypothesis.

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Counterexample: $f(x)=1/x$ on $[-1,1]$ . Endpoint signs differ, but the function is not continuous on the interval. The missing hypothesis is continuity.

Answer 14 from the source-traced unit appendix.

Full answers and selected worked solutions are in the appendices. Do not read them until you have completed an honest attempt. The universe already contains enough answer keys masquerading as education.

After the explanation

Use the section idea

Reading lens

Can you diagnose the limit type and justify a method before beginning the algebra?

Mental model

A mixed problem is a classification task before it is a calculation: direction, substitution result, structure, and required conclusion determine the route.

Decision

Name the limit type and first legal move in a margin note, then solve and check whether the conclusion matches the graph or sign behavior.

Common trap

Pattern matching without diagnosis makes similar-looking problems blur together and hides whether the error was conceptual, algebraic, or strategic.

Check yourself

You are exam-ready when you can choose a method without a section label, explain the choice, and correct a miss by naming its exact cause.

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.