Calculus I · Limits and Continuity · diagnostic

Calculus Limits Prerequisite Diagnostic

Prerequisite Diagnostic

Calculus introduces new ideas, but many wrong calculus answers are caused by old algebra. Complete this diagnostic without a calculator. A score below 15 out of 20 does not mean you cannot learn calculus. It means you should repair the identified skill before the notation becomes crowded enough to hide the original mistake.

Diagnostic Questions

Exercise 1

Factor completely: $x^2-25$ .

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$(x-5)(x+5)$ .

Answer 1 from the source-traced unit appendix.

Exercise 2

Factor completely: $x^3-8$ .

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$(x-2)(x^2+2x+4)$ .

Answer 2 from the source-traced unit appendix.

Exercise 3

Factor completely: $2x^2-5x-3$ .

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$(2x+1)(x-3)$ .

Answer 3 from the source-traced unit appendix.

Exercise 4

Simplify and state all excluded values:

\frac{x^2-9}{x-3}.

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$x+3$ , with $x\ne3$ .

Answer 4 from the source-traced unit appendix.

Exercise 5

Simplify:

\frac{\frac{1}{x+2}-\frac12}{x}.

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$-1/[2(x+2)]$ , with the relevant exclusions inherited from the original expression.

Answer 5 from the source-traced unit appendix.

Exercise 6

Rationalize the numerator:

\frac{\sqrt{x+9}-3}{x}.

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$1/(\sqrt{x+9}+3)$ .

Answer 6 from the source-traced unit appendix.

Exercise 7

Solve $|x-4|<0.3$ .

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$3.7<x<4.3$ .

Answer 7 from the source-traced unit appendix.

Exercise 8

Rewrite $\sqrt{x^2}$ correctly for every real $x$ .

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$|x|$ .

Answer 8 from the source-traced unit appendix.

Exercise 9

State the domain of $f(x)=\dfrac{1}{x^2-4}$ .

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$x\ne-2,2$ ; domain $( -\infty,-2)\cup(-2,2)\cup(2,\infty)$ .

Answer 9 from the source-traced unit appendix.

Exercise 10

State the domain of $g(x)=\sqrt{5-x}$ .

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$x\le5$ .

Answer 10 from the source-traced unit appendix.

Exercise 11

Evaluate $\sin(\pi/6)$ , $\cos(\pi/3)$ , and $\tan(\pi/4)$ .

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$1/2,1/2,1$ .

Answer 11 from the source-traced unit appendix.

Exercise 12

Simplify $1-\cos^2\theta$ .

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$\sin^2\theta$ .

Answer 12 from the source-traced unit appendix.

Exercise 13

If $f(x)=x^2-3x$ , find $f(2+h)$ .

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$f(2+h)=h^2+h-2$ .

Answer 13 from the source-traced unit appendix.

Exercise 14

For the same function, simplify

\frac{f(2+h)-f(2)}{h},\qquad h\ne0.

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$h+1$ , for $h\ne0$ .

Answer 14 from the source-traced unit appendix.

Exercise 15

Solve $3x+1=10$ .

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$x=3$ .

Answer 15 from the source-traced unit appendix.

Exercise 16

Solve $2a+1=4-a$ .

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$a=1$ .

Answer 16 from the source-traced unit appendix.

Exercise 17

Find the slope through $(-1,2)$ and $(3,10)$ .

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$2$ .

Answer 17 from the source-traced unit appendix.

Exercise 18

Explain the difference between $x=2$ and $x\to2$ .

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$x=2$ is equality; $x\to2$ describes nearby values approaching $2$ .

Answer 18 from the source-traced unit appendix.

Exercise 19

Explain why $\dfrac00$ is not equal to zero.

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Division by zero is undefined; no number multiplied by zero can recover the required numerator in a unique way.

Answer 19 from the source-traced unit appendix.

Exercise 20

Sketch any function with a hole at $(1,3)$ and a filled point at $(1,5)$ .

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Any sketch with an open circle at $(1,3)$ and a filled circle at $(1,5)$ .

Answer 20 from the source-traced unit appendix.

Answers and skill diagnoses appear in the referenced section.

After the explanation

Use the section idea

Reading lens

What information does limit notation give you, and what does it deliberately leave open?

Mental model

Treat a limit as a claim about a neighborhood around an input, not as a command to plug in the input itself.

Decision

Before calculating, identify the input target, the output target, and whether the approach is two-sided, one-sided, or toward infinity.

Common trap

Do not let a filled point, an undefined value, or unfamiliar notation distract you from the nearby behavior the limit actually describes.

Check yourself

You are ready to continue when you can read a limit aloud and name every part of its notation without evaluating it.

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.