Limits by factoring worksheet for Calculus

Substituting gives zero over zero, which is not an answer, it is a signal. The same factor is on the top and the bottom; cancel it and the limit is whatever is left. When the factor is hidden, multiply by a conjugate or clear the fraction inside the fraction; the trig ones come down to sin x/x going to 1.

Also taught in Precalculus, at its own level and with its own examples.

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Pick more than one for a sheet that mixes them.

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Sigma Prep · sigmaprep.io/worksheets Name Limits at Removable Discontinuities Date Period Evaluate each limit, or answer as asked. Evaluate limits by factoring and canceling $\lim_{x \to 1} \frac{x^{2} - x}{x - 1}$ $\lim_{x \to - 4} \frac{x^{2} + 8 x + 16}{x + 4}$ $\lim_{x \to 5} \frac{x^{2} + x - 30}{x - 5}$ Evaluate limits using special factoring patterns $\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ $\lim_{x \to - 4} \frac{x^{2} - 16}{x + 4}$ $\lim_{x \to - 1} \frac{x^{2} - 1}{x + 1}$ Evaluate limits using the conjugate $\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3}$ $\lim_{x \to - 2} \frac{\sqrt{x + 3} - 1}{x + 2}$ $\lim_{x \to 0} \frac{\sqrt{x + 9} - 3}{x}$ Evaluate limits with complex fractions $\lim_{x \to - 4} \frac{\frac{1}{x} + \frac{1}{4}}{x + 4}$ $\lim_{x \to 3} \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$ $\lim_{x \to 1} \frac{\frac{1}{x} - 1}{x - 1}$ Use $\lim_{x \to 0} \frac{\sin x}{x} = 1$ $\lim_{x \to 0} \frac{\sin 5 x}{x}$ $\lim_{x \to 0} \frac{\tan 3 x}{2 x}$ $\lim_{x \to 0} \frac{\sin 7 x}{8 x}$ Evaluate limits by factoring higher powers $\lim_{x \to 2} \frac{x^{3} - 8}{x^{2} - 4}$ $\lim_{x \to - 4} \frac{x^{3} + 64}{x + 4}$ $\lim_{x \to - 4} \frac{x^{3} + 64}{x^{2} + 4 x}$ Evaluate limits that start as $\frac{0}{0}$ $\lim_{x \to 0} \frac{\sin^{2} 3 x}{2 x^{2}}$ $\lim_{x \to - 2} \frac{x^{2} - 4}{x^{3} + 8}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Limits at Removable Discontinuities

Date Period

Evaluate each limit, or answer as asked.

Evaluate limits by factoring and canceling
$\lim_{x \to 1} \frac{x^{2} - x}{x - 1}$
$\lim_{x \to - 4} \frac{x^{2} + 8 x + 16}{x + 4}$
$\lim_{x \to 5} \frac{x^{2} + x - 30}{x - 5}$

Evaluate limits using special factoring patterns
$\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$
$\lim_{x \to - 4} \frac{x^{2} - 16}{x + 4}$
$\lim_{x \to - 1} \frac{x^{2} - 1}{x + 1}$

Evaluate limits using the conjugate
$\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3}$
$\lim_{x \to - 2} \frac{\sqrt{x + 3} - 1}{x + 2}$
$\lim_{x \to 0} \frac{\sqrt{x + 9} - 3}{x}$

Evaluate limits with complex fractions
$\lim_{x \to - 4} \frac{\frac{1}{x} + \frac{1}{4}}{x + 4}$
$\lim_{x \to 3} \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$
$\lim_{x \to 1} \frac{\frac{1}{x} - 1}{x - 1}$

Use $\lim_{x \to 0} \frac{\sin x}{x} = 1$
$\lim_{x \to 0} \frac{\sin 5 x}{x}$
$\lim_{x \to 0} \frac{\tan 3 x}{2 x}$
$\lim_{x \to 0} \frac{\sin 7 x}{8 x}$

Evaluate limits by factoring higher powers
$\lim_{x \to 2} \frac{x^{3} - 8}{x^{2} - 4}$
$\lim_{x \to - 4} \frac{x^{3} + 64}{x + 4}$
$\lim_{x \to - 4} \frac{x^{3} + 64}{x^{2} + 4 x}$

Evaluate limits that start as $\frac{0}{0}$
$\lim_{x \to 0} \frac{\sin^{2} 3 x}{2 x^{2}}$
$\lim_{x \to - 2} \frac{x^{2} - 4}{x^{3} + 8}$



Sigma Prep · sigmaprep.io/worksheets Name Limits at Removable DiscontinuitiesAnswer keyVersion 1 Date Period Evaluate each limit, or answer as asked. Evaluate limits by factoring and canceling $\lim_{x \to 1} \frac{x^{2} - x}{x - 1}$ $1$ $\lim_{x \to - 4} \frac{x^{2} + 8 x + 16}{x + 4}$ $0$ $\lim_{x \to 5} \frac{x^{2} + x - 30}{x - 5}$ $11$ Evaluate limits using special factoring patterns $\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ $6$ $\lim_{x \to - 4} \frac{x^{2} - 16}{x + 4}$ $- 8$ $\lim_{x \to - 1} \frac{x^{2} - 1}{x + 1}$ $- 2$ Evaluate limits using the conjugate $\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3}$ $\frac{1}{4}$ $\lim_{x \to - 2} \frac{\sqrt{x + 3} - 1}{x + 2}$ $\frac{1}{2}$ $\lim_{x \to 0} \frac{\sqrt{x + 9} - 3}{x}$ $\frac{1}{6}$ Evaluate limits with complex fractions $\lim_{x \to - 4} \frac{\frac{1}{x} + \frac{1}{4}}{x + 4}$ $- \frac{1}{16}$ $\lim_{x \to 3} \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$ $- \frac{1}{9}$ $\lim_{x \to 1} \frac{\frac{1}{x} - 1}{x - 1}$ $- 1$ Use $\lim_{x \to 0} \frac{\sin x}{x} = 1$ $\lim_{x \to 0} \frac{\sin 5 x}{x}$ $5$ $\lim_{x \to 0} \frac{\tan 3 x}{2 x}$ $\frac{3}{2}$ $\lim_{x \to 0} \frac{\sin 7 x}{8 x}$ $\frac{7}{8}$ Evaluate limits by factoring higher powers $\lim_{x \to 2} \frac{x^{3} - 8}{x^{2} - 4}$ $3$ $\lim_{x \to - 4} \frac{x^{3} + 64}{x + 4}$ $48$ $\lim_{x \to - 4} \frac{x^{3} + 64}{x^{2} + 4 x}$ $- 12$ Evaluate limits that start as $\frac{0}{0}$ $\lim_{x \to 0} \frac{\sin^{2} 3 x}{2 x^{2}}$ $\frac{9}{2}$ $\lim_{x \to - 2} \frac{x^{2} - 4}{x^{3} + 8}$ $- \frac{1}{3}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Limits at Removable DiscontinuitiesAnswer keyVersion 1

Date Period

Evaluate each limit, or answer as asked.

Evaluate limits by factoring and canceling
$\lim_{x \to 1} \frac{x^{2} - x}{x - 1}$ $1$
$\lim_{x \to - 4} \frac{x^{2} + 8 x + 16}{x + 4}$ $0$
$\lim_{x \to 5} \frac{x^{2} + x - 30}{x - 5}$ $11$

Evaluate limits using special factoring patterns
$\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$ $6$
$\lim_{x \to - 4} \frac{x^{2} - 16}{x + 4}$ $- 8$
$\lim_{x \to - 1} \frac{x^{2} - 1}{x + 1}$ $- 2$

Evaluate limits using the conjugate
$\lim_{x \to 3} \frac{\sqrt{x + 1} - 2}{x - 3}$ $\frac{1}{4}$
$\lim_{x \to - 2} \frac{\sqrt{x + 3} - 1}{x + 2}$ $\frac{1}{2}$
$\lim_{x \to 0} \frac{\sqrt{x + 9} - 3}{x}$ $\frac{1}{6}$

Evaluate limits with complex fractions
$\lim_{x \to - 4} \frac{\frac{1}{x} + \frac{1}{4}}{x + 4}$ $- \frac{1}{16}$
$\lim_{x \to 3} \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$ $- \frac{1}{9}$
$\lim_{x \to 1} \frac{\frac{1}{x} - 1}{x - 1}$ $- 1$

Use $\lim_{x \to 0} \frac{\sin x}{x} = 1$
$\lim_{x \to 0} \frac{\sin 5 x}{x}$ $5$
$\lim_{x \to 0} \frac{\tan 3 x}{2 x}$ $\frac{3}{2}$
$\lim_{x \to 0} \frac{\sin 7 x}{8 x}$ $\frac{7}{8}$

Evaluate limits by factoring higher powers
$\lim_{x \to 2} \frac{x^{3} - 8}{x^{2} - 4}$ $3$
$\lim_{x \to - 4} \frac{x^{3} + 64}{x + 4}$ $48$
$\lim_{x \to - 4} \frac{x^{3} + 64}{x^{2} + 4 x}$ $- 12$

Evaluate limits that start as $\frac{0}{0}$
$\lim_{x \to 0} \frac{\sin^{2} 3 x}{2 x^{2}}$ $\frac{9}{2}$
$\lim_{x \to - 2} \frac{x^{2} - 4}{x^{3} + 8}$ $- \frac{1}{3}$

How to do these

$\lim_{x \to 3} \frac{x^{2} - 9}{x - 3}$

Substituting gives 0 over 0.
The top factors as (x - 3)(x + 3).
Cancel and put 3 into x + 3.

The answer is $6$ .

Where students go wrong

Calling 0/0 zero, or undefined, and stopping. It means the answer is still hidden: the function has a hole there, not a wall, and the limit exists even though the value does not.

Also called limits by factoring, indeterminate form, zero over zero or holes.

What you can put on this worksheet

A factor that cancels: $\lim_{x \to 3} \frac{x^{2} - 7 x + 12}{x - 3}$ → $- 1$
Difference of squares and cubes: $\lim_{x \to 5} \frac{x^{2} - 25}{x - 5}$ → $10$
Multiplying by the conjugate: $\lim_{x \to 2} \frac{\sqrt{x + 2} - 2}{x - 2}$ → $\frac{1}{4}$
Fractions inside fractions: $\lim_{x \to 3} \frac{\frac{1}{x} - \frac{1}{3}}{x - 3}$ → $- \frac{1}{9}$
The sin x / x family: $\lim_{x \to 0} \frac{\tan 6 x}{6 x}$ → $1$
A cubic or quartic to factor, or grouping: $\lim_{x \to 3} \frac{x - 3}{x^{3} - 27}$ → $\frac{1}{27}$
Squared and cofunction trig, a stated point value, a fraction below: $\lim_{x \to - 3} \frac{x^{2} + 2 x - 3}{x + 3}$ → $- 4$

Questions about these worksheets

Yes. Every limits by factoring sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the limits by factoring questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 7: a factor that cancels, difference of squares and cubes, multiplying by the conjugate, fractions inside fractions, the sin x / x family, a cubic or quartic to factor, or grouping and squared and cofunction trig, a stated point value, a fraction below. Tick as many as you want and set how many of each, or let it spread them evenly.

Calling 0/0 zero, or undefined, and stopping. It means the answer is still hidden: the function has a hole there, not a wall, and the limit exists even though the value does not.

Limits by factoring worksheet for Calculus

Select your difficulty

Evaluate limits by factoring and canceling

Evaluate limits using special factoring patterns

Evaluate limits using the conjugate

Evaluate limits with complex fractions

Use lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1limx→0​xsinx​=1

Evaluate limits by factoring higher powers

Evaluate limits that start as 00\frac{0}{0}00​

Evaluate limits by factoring and canceling

Evaluate limits using special factoring patterns

Evaluate limits using the conjugate

Evaluate limits with complex fractions

Use lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1limx→0​xsinx​=1

Evaluate limits by factoring higher powers

Evaluate limits that start as 00\frac{0}{0}00​

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these limits by factoring worksheets free?+

Do the limits by factoring worksheets come with an answer key?+

Are these limits by factoring worksheets printable?+

What grade are these limits by factoring worksheets for?+

Can I get a different version of the same limits by factoring worksheet?+

Is this a free alternative to Kuta Software for limits by factoring?+

Can I choose how hard the limits by factoring questions are?+

What kinds of question are on a limits by factoring worksheet?+

What do students get wrong with limits by factoring?+

Use $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Evaluate limits that start as $\frac{0}{0}$

Use $\lim_{x \to 0} \frac{\sin x}{x} = 1$

Evaluate limits that start as $\frac{0}{0}$