Calculus I · Limits and Continuity · diagnostic

Calculus Limits Prerequisite Diagnostic

Check the algebra and function skills needed for limits and continuity, then follow targeted review links before starting Calculus I.

Course progress2 of 47 core pages2 of 47

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

Factor completely: $x^2-25$ .

Exercise

Factor completely: $x^3-8$ .

Exercise

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

Exercise

Simplify and state all excluded values:

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

Exercise

Simplify:

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

Exercise

Rationalize the numerator:

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

Exercise

Solve $|x-4|<0.3$ .

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

For the same function, simplify

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

Exercise

Solve $3x+1=10$ .

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Answers and skill diagnoses appear in the referenced section.

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.

Calculus I · Limits and Continuity · diagnostic

Calculus Limits Prerequisite Diagnostic

Check the algebra and function skills needed for limits and continuity, then follow targeted review links before starting Calculus I.

Course progress2 of 47 core pages2 of 47

Prerequisite Diagnostic

Diagnostic Questions

Exercise

Factor completely: $x^2-25$ .

Exercise

Factor completely: $x^3-8$ .

Exercise

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

Exercise

Simplify and state all excluded values:

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

Exercise

Simplify:

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

Exercise

Rationalize the numerator:

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

Exercise

Solve $|x-4|<0.3$ .

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

For the same function, simplify

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

Exercise

Solve $3x+1=10$ .

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Exercise

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

Answers and skill diagnoses appear in the referenced section.

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.

Prerequisite Diagnostic

Diagnostic Questions

Related resources

Original instruction with traceable references.

Prerequisite Diagnostic

Diagnostic Questions

Related resources

Original instruction with traceable references.