Worksheets · Precalculus

Precalculus one-sided limits worksheet

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.

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Name

One‑Sided Limits

Date Period

Evaluate each limit.

  1. lim⁡x→1+f(x) for f(x)={6x+9x<1−4x+4x≥1
  2. lim⁡x→−3+f(x) for f(x)={−3x−5x<−3−3x2−8x≥−3
  3. lim⁡x→−2+f(x) for f(x)={−3x+2x<−22x−6x≥−2
  4. lim⁡x→2−f(x) for f(x)={3x+5x<22x2+5x≥2
  5. lim⁡x→−2+f(x) for f(x)={−5x+1x<−2−2x−9x≥−2
  6. lim⁡x→−6+f(x) for f(x)={3x−4x<−66x+8x≥−6
  7. lim⁡x→6+f(x) for f(x)={−x+7x<6−6xx≥6
  8. lim⁡x→−5−f(x) for f(x)={x+9x<−5−3x−7x≥−5
  9. lim⁡x→4+f(x) for f(x)={x+6x<42x2−6x≥4
  10. lim⁡x→−1+f(x) for f(x)={4x−3x<−1−x2−8x≥−1
  11. lim⁡x→1+f(x) for f(x)={4x−5x<12x+9x≥1
  12. lim⁡x→−2+f(x) for f(x)={−6x−3x<−26x2−8x≥−2
  13. lim⁡x→5−f(x) for f(x)={−x−7x<5−4x−4x≥5
  14. lim⁡x→6+f(x) for f(x)={5x−5x<6−2x2−6x≥6
  15. lim⁡x→1+f(x) for f(x)={−x−9x<13x2−5x≥1
  16. lim⁡x→−6+f(x) for f(x)={3x+4x<−6−2x+1x≥−6
  17. lim⁡x→−2−f(x) for f(x)={4x−9x<−2−3x+7x≥−2
  18. lim⁡x→−4+f(x) for f(x)={−6x+1x<−44x2−7x≥−4
  19. lim⁡x→−5−f(x) for f(x)={−4x−4x<−56x2−8x≥−5
  20. lim⁡x→1−f(x) for f(x)={3x+5x<1−x+9x≥1

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.

  1. lim⁡x→1+f(x) for f(x)={6x+9x<1−4x+4x≥10
  2. lim⁡x→−3+f(x) for f(x)={−3x−5x<−3−3x2−8x≥−3−35
  3. lim⁡x→−2+f(x) for f(x)={−3x+2x<−22x−6x≥−2−10
  4. lim⁡x→2−f(x) for f(x)={3x+5x<22x2+5x≥211
  5. lim⁡x→−2+f(x) for f(x)={−5x+1x<−2−2x−9x≥−2−5
  6. lim⁡x→−6+f(x) for f(x)={3x−4x<−66x+8x≥−6−28
  7. lim⁡x→6+f(x) for f(x)={−x+7x<6−6xx≥6−36
  8. lim⁡x→−5−f(x) for f(x)={x+9x<−5−3x−7x≥−54
  9. lim⁡x→4+f(x) for f(x)={x+6x<42x2−6x≥426
  10. lim⁡x→−1+f(x) for f(x)={4x−3x<−1−x2−8x≥−1−9
  11. lim⁡x→1+f(x) for f(x)={4x−5x<12x+9x≥111
  12. lim⁡x→−2+f(x) for f(x)={−6x−3x<−26x2−8x≥−216
  13. lim⁡x→5−f(x) for f(x)={−x−7x<5−4x−4x≥5−12
  14. lim⁡x→6+f(x) for f(x)={5x−5x<6−2x2−6x≥6−78
  15. lim⁡x→1+f(x) for f(x)={−x−9x<13x2−5x≥1−2
  16. lim⁡x→−6+f(x) for f(x)={3x+4x<−6−2x+1x≥−613
  17. lim⁡x→−2−f(x) for f(x)={4x−9x<−2−3x+7x≥−2−17
  18. lim⁡x→−4+f(x) for f(x)={−6x+1x<−44x2−7x≥−457
  19. lim⁡x→−5−f(x) for f(x)={−4x−4x<−56x2−8x≥−516
  20. lim⁡x→1−f(x) for f(x)={3x+5x<1−x+9x≥18

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

How to do these

lim⁡x→2−f(x) for f(x)={3x−1x<2x2x≥2

  1. From the left means x is below 2.
  2. So the first rule applies.
  3. 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→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

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 2: one-sided limits and whether the limit exists. 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.