One-sided limits worksheet for Calculus

Q: Are these one-sided limits worksheets free?

Yes. Every one-sided limits 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.

Q: Do the one-sided limits worksheets come with an answer key?

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.

Q: Are these one-sided limits worksheets printable?

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.

Q: What grade are these one-sided limits worksheets for?

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.

Q: Can I get a different version of the same one-sided limits worksheet?

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.

Q: Is this a free alternative to Kuta Software for one-sided limits?

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 one-sided limits 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.

Q: Can I choose how hard the one-sided limits questions are?

Yes. There are three 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.

Q: What kinds of question are on a one-sided limits worksheet?

You pick from 4: one-sided limits, whether the limit exists, absolute value over x - a and trinomial and constant pieces, and a factor over an absolute value. Tick as many as you want and set how many of each, or let it spread them evenly.

Q: What do students get wrong with one-sided limits?

Using the rule that owns the break point. Coming in from the left you never reach it, so the condition at x itself does not decide anything: the side you are on does.

A little minus sign means coming in from the left and a plus from the right. Which rule of a piecewise function applies depends on which side you are on, and the two-sided limit only exists when both give the same answer.

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

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name One-Sided Limits Date Period Evaluate each limit, or answer as asked. Find one-sided limits of piecewise functions $\lim_{x \to 1^{+}} f (x)$ for $f (x) = {\begin{cases} 5 x + 9 & x < 1 \\ - 4 x + 4 & x \geq 1 \end{cases}$ $\lim_{x \to 2^{+}} f (x)$ for $f (x) = {\begin{cases} 4 x - 2 & x < 2 \\ - 4 x + 9 & x \geq 2 \end{cases}$ $\lim_{x \to - 1^{+}} f (x)$ for $f (x) = {\begin{cases} 2 x + 4 & x < - 1 \\ x - 5 & x \geq - 1 \end{cases}$ $\lim_{x \to - 5^{-}} f (x)$ for $f (x) = {\begin{cases} - 4 x - 4 & x < - 5 \\ 2 x + 1 & x \geq - 5 \end{cases}$ $\lim_{x \to 2^{+}} f (x)$ for $f (x) = {\begin{cases} - 4 x + 2 & x < 2 \\ - x - 9 & x \geq 2 \end{cases}$ $\lim_{x \to 3^{+}} f (x)$ for $f (x) = {\begin{cases} - 4 x + 1 & x < 3 \\ 5 x - 2 & x \geq 3 \end{cases}$ $\lim_{x \to 4^{-}} f (x)$ for $f (x) = {\begin{cases} - x - 6 & x < 4 \\ 3 x & x \geq 4 \end{cases}$ Evaluate limits of absolute value quotients $\lim_{x \to - 1^{-}} \frac{2 ⁣ ∣ x + 1 ∣}{x + 1}$ $\lim_{x \to 5} \frac{4 ⁣ ∣ x - 5 ∣}{x - 5}$ $\lim_{x \to 6^{-}} \frac{4 ⁣ ∣ x - 6 ∣}{x - 6}$ $\lim_{x \to - 5} \frac{∣ x + 5 ∣}{x + 5}$ $\lim_{x \to 4^{+}} \frac{3 ⁣ ∣ x - 4 ∣}{x - 4}$ $\lim_{x \to - 6} \frac{2 ⁣ ∣ x + 6 ∣}{x + 6}$ $\lim_{x \to 0} \frac{2 ⁣ ∣ x ∣}{x}$ Find one-sided limits Find $\lim_{x \to 1^{+}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x + 2, & x < 1 \\ 5, & x \geq 1 \end{cases}$ Find $\lim_{x \to - 1^{-}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x - 3, & x < - 1 \\ 3, & x \geq - 1 \end{cases}$ Find $\lim_{x \to - 1^{+}} f (x)$ for $f (x) = {\begin{cases} 2 x^{2} - 5 x + 2, & x < - 1 \\ 4 x - 5, & x \geq - 1 \end{cases}$ Find $\lim_{x \to - 1^{-}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x + 1, & x < - 1 \\ - 2, & x \geq - 1 \end{cases}$ Find $\lim_{x \to - 2^{-}} f (x)$ for $f (x) = {\begin{cases} - x^{2} + 5 x - 7, & x < - 2 \\ - 2 x + 6, & x \geq - 2 \end{cases}$ Find $\lim_{x \to - 3^{-}} f (x)$ for $f (x) = {\begin{cases} 2 x^{2} - 5 x - 4, & x < - 3 \\ 3 x + 4, & x \geq - 3 \end{cases}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

One-Sided Limits

Date Period

Evaluate each limit, or answer as asked.

Find one-sided limits of piecewise functions
$\lim_{x \to 1^{+}} f (x)$ for $f (x) = {\begin{cases} 5 x + 9 & x < 1 \\ - 4 x + 4 & x \geq 1 \end{cases}$
$\lim_{x \to 2^{+}} f (x)$ for $f (x) = {\begin{cases} 4 x - 2 & x < 2 \\ - 4 x + 9 & x \geq 2 \end{cases}$
$\lim_{x \to - 1^{+}} f (x)$ for $f (x) = {\begin{cases} 2 x + 4 & x < - 1 \\ x - 5 & x \geq - 1 \end{cases}$
$\lim_{x \to - 5^{-}} f (x)$ for $f (x) = {\begin{cases} - 4 x - 4 & x < - 5 \\ 2 x + 1 & x \geq - 5 \end{cases}$
$\lim_{x \to 2^{+}} f (x)$ for $f (x) = {\begin{cases} - 4 x + 2 & x < 2 \\ - x - 9 & x \geq 2 \end{cases}$
$\lim_{x \to 3^{+}} f (x)$ for $f (x) = {\begin{cases} - 4 x + 1 & x < 3 \\ 5 x - 2 & x \geq 3 \end{cases}$
$\lim_{x \to 4^{-}} f (x)$ for $f (x) = {\begin{cases} - x - 6 & x < 4 \\ 3 x & x \geq 4 \end{cases}$

Evaluate limits of absolute value quotients
$\lim_{x \to - 1^{-}} \frac{2 ⁣ ∣ x + 1 ∣}{x + 1}$
$\lim_{x \to 5} \frac{4 ⁣ ∣ x - 5 ∣}{x - 5}$
$\lim_{x \to 6^{-}} \frac{4 ⁣ ∣ x - 6 ∣}{x - 6}$
$\lim_{x \to - 5} \frac{∣ x + 5 ∣}{x + 5}$
$\lim_{x \to 4^{+}} \frac{3 ⁣ ∣ x - 4 ∣}{x - 4}$
$\lim_{x \to - 6} \frac{2 ⁣ ∣ x + 6 ∣}{x + 6}$
$\lim_{x \to 0} \frac{2 ⁣ ∣ x ∣}{x}$

Find one-sided limits
Find $\lim_{x \to 1^{+}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x + 2, & x < 1 \\ 5, & x \geq 1 \end{cases}$
Find $\lim_{x \to - 1^{-}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x - 3, & x < - 1 \\ 3, & x \geq - 1 \end{cases}$
Find $\lim_{x \to - 1^{+}} f (x)$ for $f (x) = {\begin{cases} 2 x^{2} - 5 x + 2, & x < - 1 \\ 4 x - 5, & x \geq - 1 \end{cases}$
Find $\lim_{x \to - 1^{-}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x + 1, & x < - 1 \\ - 2, & x \geq - 1 \end{cases}$
Find $\lim_{x \to - 2^{-}} f (x)$ for $f (x) = {\begin{cases} - x^{2} + 5 x - 7, & x < - 2 \\ - 2 x + 6, & x \geq - 2 \end{cases}$
Find $\lim_{x \to - 3^{-}} f (x)$ for $f (x) = {\begin{cases} 2 x^{2} - 5 x - 4, & x < - 3 \\ 3 x + 4, & x \geq - 3 \end{cases}$



Sigma Prep · sigmaprep.io/worksheets Name One-Sided LimitsAnswer keyVersion 1 Date Period Evaluate each limit, or answer as asked. Find one-sided limits of piecewise functions $\lim_{x \to 1^{+}} f (x)$ for $f (x) = {\begin{cases} 5 x + 9 & x < 1 \\ - 4 x + 4 & x \geq 1 \end{cases}$ $0$ $\lim_{x \to 2^{+}} f (x)$ for $f (x) = {\begin{cases} 4 x - 2 & x < 2 \\ - 4 x + 9 & x \geq 2 \end{cases}$ $1$ $\lim_{x \to - 1^{+}} f (x)$ for $f (x) = {\begin{cases} 2 x + 4 & x < - 1 \\ x - 5 & x \geq - 1 \end{cases}$ $- 6$ $\lim_{x \to - 5^{-}} f (x)$ for $f (x) = {\begin{cases} - 4 x - 4 & x < - 5 \\ 2 x + 1 & x \geq - 5 \end{cases}$ $16$ $\lim_{x \to 2^{+}} f (x)$ for $f (x) = {\begin{cases} - 4 x + 2 & x < 2 \\ - x - 9 & x \geq 2 \end{cases}$ $- 11$ $\lim_{x \to 3^{+}} f (x)$ for $f (x) = {\begin{cases} - 4 x + 1 & x < 3 \\ 5 x - 2 & x \geq 3 \end{cases}$ $13$ $\lim_{x \to 4^{-}} f (x)$ for $f (x) = {\begin{cases} - x - 6 & x < 4 \\ 3 x & x \geq 4 \end{cases}$ $- 10$ Evaluate limits of absolute value quotients $\lim_{x \to - 1^{-}} \frac{2 ⁣ ∣ x + 1 ∣}{x + 1}$ $- 2$ $\lim_{x \to 5} \frac{4 ⁣ ∣ x - 5 ∣}{x - 5}$ Does not exist $\lim_{x \to 6^{-}} \frac{4 ⁣ ∣ x - 6 ∣}{x - 6}$ $- 4$ $\lim_{x \to - 5} \frac{∣ x + 5 ∣}{x + 5}$ Does not exist $\lim_{x \to 4^{+}} \frac{3 ⁣ ∣ x - 4 ∣}{x - 4}$ $3$ $\lim_{x \to - 6} \frac{2 ⁣ ∣ x + 6 ∣}{x + 6}$ Does not exist $\lim_{x \to 0} \frac{2 ⁣ ∣ x ∣}{x}$ Does not exist Find one-sided limits Find $\lim_{x \to 1^{+}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x + 2, & x < 1 \\ 5, & x \geq 1 \end{cases}$ $5$ Find $\lim_{x \to - 1^{-}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x - 3, & x < - 1 \\ 3, & x \geq - 1 \end{cases}$ $- \frac{7}{2}$ Find $\lim_{x \to - 1^{+}} f (x)$ for $f (x) = {\begin{cases} 2 x^{2} - 5 x + 2, & x < - 1 \\ 4 x - 5, & x \geq - 1 \end{cases}$ $- 9$ Find $\lim_{x \to - 1^{-}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x + 1, & x < - 1 \\ - 2, & x \geq - 1 \end{cases}$ $\frac{1}{2}$ Find $\lim_{x \to - 2^{-}} f (x)$ for $f (x) = {\begin{cases} - x^{2} + 5 x - 7, & x < - 2 \\ - 2 x + 6, & x \geq - 2 \end{cases}$ $- 21$ Find $\lim_{x \to - 3^{-}} f (x)$ for $f (x) = {\begin{cases} 2 x^{2} - 5 x - 4, & x < - 3 \\ 3 x + 4, & x \geq - 3 \end{cases}$ $29$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

One-Sided LimitsAnswer keyVersion 1

Date Period

Evaluate each limit, or answer as asked.

Find one-sided limits of piecewise functions
$\lim_{x \to 1^{+}} f (x)$ for $f (x) = {\begin{cases} 5 x + 9 & x < 1 \\ - 4 x + 4 & x \geq 1 \end{cases}$ $0$
$\lim_{x \to 2^{+}} f (x)$ for $f (x) = {\begin{cases} 4 x - 2 & x < 2 \\ - 4 x + 9 & x \geq 2 \end{cases}$ $1$
$\lim_{x \to - 1^{+}} f (x)$ for $f (x) = {\begin{cases} 2 x + 4 & x < - 1 \\ x - 5 & x \geq - 1 \end{cases}$ $- 6$
$\lim_{x \to - 5^{-}} f (x)$ for $f (x) = {\begin{cases} - 4 x - 4 & x < - 5 \\ 2 x + 1 & x \geq - 5 \end{cases}$ $16$
$\lim_{x \to 2^{+}} f (x)$ for $f (x) = {\begin{cases} - 4 x + 2 & x < 2 \\ - x - 9 & x \geq 2 \end{cases}$ $- 11$
$\lim_{x \to 3^{+}} f (x)$ for $f (x) = {\begin{cases} - 4 x + 1 & x < 3 \\ 5 x - 2 & x \geq 3 \end{cases}$ $13$
$\lim_{x \to 4^{-}} f (x)$ for $f (x) = {\begin{cases} - x - 6 & x < 4 \\ 3 x & x \geq 4 \end{cases}$ $- 10$

Evaluate limits of absolute value quotients
$\lim_{x \to - 1^{-}} \frac{2 ⁣ ∣ x + 1 ∣}{x + 1}$ $- 2$
$\lim_{x \to 5} \frac{4 ⁣ ∣ x - 5 ∣}{x - 5}$ Does not exist
$\lim_{x \to 6^{-}} \frac{4 ⁣ ∣ x - 6 ∣}{x - 6}$ $- 4$
$\lim_{x \to - 5} \frac{∣ x + 5 ∣}{x + 5}$ Does not exist
$\lim_{x \to 4^{+}} \frac{3 ⁣ ∣ x - 4 ∣}{x - 4}$ $3$
$\lim_{x \to - 6} \frac{2 ⁣ ∣ x + 6 ∣}{x + 6}$ Does not exist
$\lim_{x \to 0} \frac{2 ⁣ ∣ x ∣}{x}$ Does not exist

Find one-sided limits
Find $\lim_{x \to 1^{+}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x + 2, & x < 1 \\ 5, & x \geq 1 \end{cases}$ $5$
Find $\lim_{x \to - 1^{-}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x - 3, & x < - 1 \\ 3, & x \geq - 1 \end{cases}$ $- \frac{7}{2}$
Find $\lim_{x \to - 1^{+}} f (x)$ for $f (x) = {\begin{cases} 2 x^{2} - 5 x + 2, & x < - 1 \\ 4 x - 5, & x \geq - 1 \end{cases}$ $- 9$
Find $\lim_{x \to - 1^{-}} f (x)$ for $f (x) = {\begin{cases} \frac{1}{2} x + 1, & x < - 1 \\ - 2, & x \geq - 1 \end{cases}$ $\frac{1}{2}$
Find $\lim_{x \to - 2^{-}} f (x)$ for $f (x) = {\begin{cases} - x^{2} + 5 x - 7, & x < - 2 \\ - 2 x + 6, & x \geq - 2 \end{cases}$ $- 21$
Find $\lim_{x \to - 3^{-}} f (x)$ for $f (x) = {\begin{cases} 2 x^{2} - 5 x - 4, & x < - 3 \\ 3 x + 4, & x \geq - 3 \end{cases}$ $29$

How to do these

$\lim_{x \to 2^{-}} f (x)$ for $f (x) = {\begin{cases} 3 x - 1 & x < 2 \\ x^{2} & x \geq 2 \end{cases}$

From the left means x is below 2.
So the first rule applies.
Three times 2 minus 1.

The answer is $5$ .

Where students go wrong

Using the rule that owns the break point. Coming in from the left you never reach it, so the condition at x itself does not decide anything: the side you are on does.

Also called left and right limits, piecewise limits or jump discontinuity.

What you can put on this worksheet

One-sided limits: $\lim_{x \to 2^{-}} f (x)$ for $f (x) = {\begin{cases} x & x < 2 \\ x + 9 & x \geq 2 \end{cases}$ → $2$
Whether the limit exists: Does $\lim_{x \to 2} f (x)$ exist? $f (x) = {\begin{cases} x & x < 2 \\ 3 x & x \geq 2 \end{cases}$ → No
Absolute value over x - a: $\lim_{x \to 0^{+}} \frac{∣ x ∣}{x}$ → $1$
Trinomial and constant pieces, and a factor over an absolute value: Find $\lim_{x \to 2^{-}} f (x)$ for $f (x) = {\begin{cases} x^{2} - 8, & x < 2 \\ 2 x + 5, & x \geq 2 \end{cases}$ → $- 4$

Questions about these worksheets