Worksheets · Precalculus

One-sided limits worksheet for Precalculus

Precalculus uses one-sided limits to decide whether a limit exists at a break point and to describe jumps in a graph. Expect piecewise functions where you compare the left and right limits, and quotients like |x - a|/(x - a), which give -1 on one side and 1 on the other.

Also taught in Calculus, 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.

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.

  1. Find one-sided limits of piecewise functions

    lim⁡x→2−f(x) for f(x)={−4x+7x<23x−3x≥2
  2. lim⁡x→−6+f(x) for f(x)={x+9x<−6−6x2+1x≥−6
  3. lim⁡x→−6+f(x) for f(x)={3x−3x<−6−6x−4x≥−6
  4. lim⁡x→−2+f(x) for f(x)={3x+1x<−22x−5x≥−2
  5. lim⁡x→−3−f(x) for f(x)={4x+5x<−3−5x−8x≥−3
  6. lim⁡x→5−f(x) for f(x)={6x−3x<5−4x−4x≥5
  7. lim⁡x→3+f(x) for f(x)={−4x+3x<3−6x2−2x≥3

  1. Evaluate limits of absolute value quotients

    lim⁡x→4−3 ⁣∣x−4∣x−4
  2. lim⁡x→−64 ⁣∣x+6∣x+6
  3. lim⁡x→5+4 ⁣∣x−5∣x−5
  4. lim⁡x→6∣x−6∣x−6
  5. lim⁡x→−3∣x+3∣x+3
  6. lim⁡x→0∣x∣x
  7. lim⁡x→52 ⁣∣x−5∣x−5

  1. Find one-sided limits

    lim⁡x→1+2x−2∣x−1∣
  2. Find lim⁡x→3+f(x) for f(x)={12x−6,x<36,x≥3
  3. Find lim⁡x→1−f(x) for f(x)={12x+5,x<1−7,x≥1
  4. Find lim⁡x→−1−f(x) for f(x)={12x−3,x<−1−7,x≥−1
  5. Find lim⁡x→3−f(x) for f(x)={12x+3,x<35,x≥3
  6. Find lim⁡x→1−f(x) for f(x)={12x−2,x<1−1,x≥1

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

One-Sided LimitsAnswer keyVersion 2

Date Period

Evaluate each limit, or answer as asked.

  1. Find one-sided limits of piecewise functions

    lim⁡x→2−f(x) for f(x)={−4x+7x<23x−3x≥2−1
  2. lim⁡x→−6+f(x) for f(x)={x+9x<−6−6x2+1x≥−6−215
  3. lim⁡x→−6+f(x) for f(x)={3x−3x<−6−6x−4x≥−632
  4. lim⁡x→−2+f(x) for f(x)={3x+1x<−22x−5x≥−2−9
  5. lim⁡x→−3−f(x) for f(x)={4x+5x<−3−5x−8x≥−3−7
  6. lim⁡x→5−f(x) for f(x)={6x−3x<5−4x−4x≥527
  7. lim⁡x→3+f(x) for f(x)={−4x+3x<3−6x2−2x≥3−56

  1. Evaluate limits of absolute value quotients

    lim⁡x→4−3 ⁣∣x−4∣x−4−3
  2. lim⁡x→−64 ⁣∣x+6∣x+6Does not exist
  3. lim⁡x→5+4 ⁣∣x−5∣x−54
  4. lim⁡x→6∣x−6∣x−6Does not exist
  5. lim⁡x→−3∣x+3∣x+3Does not exist
  6. lim⁡x→0∣x∣xDoes not exist
  7. lim⁡x→52 ⁣∣x−5∣x−5Does not exist

  1. Find one-sided limits

    lim⁡x→1+2x−2∣x−1∣2
  2. Find lim⁡x→3+f(x) for f(x)={12x−6,x<36,x≥36
  3. Find lim⁡x→1−f(x) for f(x)={12x+5,x<1−7,x≥1112
  4. Find lim⁡x→−1−f(x) for f(x)={12x−3,x<−1−7,x≥−1−72
  5. Find lim⁡x→3−f(x) for f(x)={12x+3,x<35,x≥392
  6. Find lim⁡x→1−f(x) for f(x)={12x−2,x<1−1,x≥1−32

Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

lim⁡x→1−f(x) for f(x)={x2+2x<14x+1x≥1

  1. From the left only inputs below 1 count, so use x² + 2.
  2. Put in 1: 1 + 2 = 3.

The answer is 3.

Where students go wrong

Using the piece that owns the point. For x → 1⁻ only inputs just below 1 matter, so the x < 1 piece gives the limit, 3, even though f(1) = 5 comes from the other piece.

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

What you can put on this worksheet

One-sided limits
lim⁡x→2−f(x) for f(x)={xx<2x+9x≥2 → 2
Whether the limit exists
Does lim⁡x→2f(x) exist? f(x)={xx<23xx≥2 → No
Absolute value over x - a
lim⁡x→0+∣x∣x → 1
Trinomial and constant pieces, and a factor over an absolute value
Find lim⁡x→2−f(x) for f(x)={x2−8,x<22x+5,x≥2 → −4

Questions about these worksheets

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.

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.

Precalculus is usually taken around 11th or 12th grade, 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 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.

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.

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.

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.