Worksheets · Calculus

Limits in the form of a derivative worksheet

Some limits are not worth taking, because they are the definition of a derivative with the function left for you to spot. Once you see which function it is, differentiating answers it in one line.

Select your difficulty

Pick more than one for a sheet that mixes them.

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

Sigma Prep · sigmaprep.io/worksheets

Name

Limits in the Form of a Derivative

Date Period

Evaluate each limit.

  1. lim⁡h→0(4+h)4−256h
  2. lim⁡h→02(3+h)4−162h
  3. lim⁡h→0−6(2+h)3+48h
  4. lim⁡h→0−3(1+h)2+3h
  5. lim⁡h→0−(4+h)3+64h
  6. lim⁡h→0−2(2+h)2+8h
  7. lim⁡h→05(1+h)3−5h
  8. lim⁡h→04(1+h)5−4h
  9. lim⁡h→03(4+h)5−3072h
  10. lim⁡h→0−5(2+h)2+20h
  11. lim⁡h→0−2(3+h)2+18h
  12. lim⁡h→0(1+h)3−1h
  13. lim⁡h→06(4+h)2−96h
  14. lim⁡h→03(3+h)3−81h
  15. lim⁡h→06(4+h)4−1536h
  16. lim⁡h→06(1+h)2−6h
  17. lim⁡h→02(3+h)5−486h
  18. lim⁡h→05(3+h)2−45h
  19. lim⁡h→0(1+h)5−1h
  20. lim⁡h→0−(2+h)3+8h

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

Sigma Prep · sigmaprep.io/worksheets

Name

Limits in the Form of a DerivativeAnswer keyVersion 1

Date Period

Evaluate each limit.

  1. lim⁡h→0(4+h)4−256h256
  2. lim⁡h→02(3+h)4−162h216
  3. lim⁡h→0−6(2+h)3+48h−72
  4. lim⁡h→0−3(1+h)2+3h−6
  5. lim⁡h→0−(4+h)3+64h−48
  6. lim⁡h→0−2(2+h)2+8h−8
  7. lim⁡h→05(1+h)3−5h15
  8. lim⁡h→04(1+h)5−4h20
  9. lim⁡h→03(4+h)5−3072h3840
  10. lim⁡h→0−5(2+h)2+20h−20
  11. lim⁡h→0−2(3+h)2+18h−12
  12. lim⁡h→0(1+h)3−1h3
  13. lim⁡h→06(4+h)2−96h48
  14. lim⁡h→03(3+h)3−81h81
  15. lim⁡h→06(4+h)4−1536h1536
  16. lim⁡h→06(1+h)2−6h12
  17. lim⁡h→02(3+h)5−486h810
  18. lim⁡h→05(3+h)2−45h30
  19. lim⁡h→0(1+h)5−1h5
  20. lim⁡h→0−(2+h)3+8h−12

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

How to do these

lim⁡h→0(x+h)4−x4h

  1. This is the definition of the derivative for f of x equals x to the fourth.
  2. So the answer is that derivative.
  3. Which is 4x cubed.

The answer is 4x3.

Where students go wrong

Expanding the bracket and grinding it out. It works, and on the fifth power it takes half a page; recognising the form turns the whole thing into one differentiation.

Also called definition of the derivative backwards, recognising the difference quotient or limit definition in reverse.

Questions about these worksheets

Yes. Every limits in the form of a derivative 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 in the form of a derivative 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.

Expanding the bracket and grinding it out. It works, and on the fifth power it takes half a page; recognising the form turns the whole thing into one differentiation.