L'Hopital's rule worksheet

When a limit comes out as 0/0 or infinity over infinity the fraction is telling you nothing yet. Differentiating the top and the bottom separately, which is not the quotient rule, usually turns it into a limit you can read straight off.

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Sigma Prep · sigmaprep.io/worksheets Name L'Hopital's Rule Date Period Evaluate each limit. $\lim_{x \to 0} \frac{e^{x} - 1}{6 x}$ $\lim_{x \to 0} \frac{e^{- 4 x} - 1}{3 x}$ $\lim_{x \to 0} \frac{\sin (- 3 x)}{- 3 x}$ $\lim_{x \to 0} \frac{e^{- 3 x} - 1}{- 3 x}$ $\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 3 x}$ $\lim_{x \to 0} \frac{e^{2 x} - 1}{- 4 x}$ $\lim_{x \to 0} \frac{e^{2 x} - 1}{3 x}$ $\lim_{x \to 0} \frac{\sin (2 x)}{x}$ $\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 5 x}$ $\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 4 x}$ $\lim_{x \to 0} \frac{\sin (- 6 x)}{3 x}$ $\lim_{x \to 0} \frac{e^{6 x} - 1}{5 x}$ $\lim_{x \to 0} \frac{e^{6 x} - 1}{- x}$ $\lim_{x \to 0} \frac{e^{- 6 x} - 1}{6 x}$ $\lim_{x \to 0} \frac{e^{- 5 x} - 1}{x}$ $\lim_{x \to 0} \frac{e^{4 x} - 1}{x}$ $\lim_{x \to 0} \frac{e^{2 x} - 1}{- 2 x}$ $\lim_{x \to 0} \frac{\sin (- x)}{4 x}$ $\lim_{x \to 0} \frac{e^{- x} - 1}{- 2 x}$ $\lim_{x \to 0} \frac{\sin (x)}{4 x}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

L'Hopital's Rule

Date Period

Evaluate each limit.

$\lim_{x \to 0} \frac{e^{x} - 1}{6 x}$
$\lim_{x \to 0} \frac{e^{- 4 x} - 1}{3 x}$
$\lim_{x \to 0} \frac{\sin (- 3 x)}{- 3 x}$
$\lim_{x \to 0} \frac{e^{- 3 x} - 1}{- 3 x}$
$\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 3 x}$
$\lim_{x \to 0} \frac{e^{2 x} - 1}{- 4 x}$
$\lim_{x \to 0} \frac{e^{2 x} - 1}{3 x}$
$\lim_{x \to 0} \frac{\sin (2 x)}{x}$
$\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 5 x}$
$\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 4 x}$
$\lim_{x \to 0} \frac{\sin (- 6 x)}{3 x}$
$\lim_{x \to 0} \frac{e^{6 x} - 1}{5 x}$
$\lim_{x \to 0} \frac{e^{6 x} - 1}{- x}$
$\lim_{x \to 0} \frac{e^{- 6 x} - 1}{6 x}$
$\lim_{x \to 0} \frac{e^{- 5 x} - 1}{x}$
$\lim_{x \to 0} \frac{e^{4 x} - 1}{x}$
$\lim_{x \to 0} \frac{e^{2 x} - 1}{- 2 x}$
$\lim_{x \to 0} \frac{\sin (- x)}{4 x}$
$\lim_{x \to 0} \frac{e^{- x} - 1}{- 2 x}$
$\lim_{x \to 0} \frac{\sin (x)}{4 x}$



Sigma Prep · sigmaprep.io/worksheets Name L'Hopital's RuleAnswer keyVersion 1 Date Period Evaluate each limit. $\lim_{x \to 0} \frac{e^{x} - 1}{6 x}$ $\frac{1}{6}$ $\lim_{x \to 0} \frac{e^{- 4 x} - 1}{3 x}$ $- \frac{4}{3}$ $\lim_{x \to 0} \frac{\sin (- 3 x)}{- 3 x}$ $1$ $\lim_{x \to 0} \frac{e^{- 3 x} - 1}{- 3 x}$ $1$ $\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 3 x}$ $\frac{2}{3}$ $\lim_{x \to 0} \frac{e^{2 x} - 1}{- 4 x}$ $- \frac{1}{2}$ $\lim_{x \to 0} \frac{e^{2 x} - 1}{3 x}$ $\frac{2}{3}$ $\lim_{x \to 0} \frac{\sin (2 x)}{x}$ $2$ $\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 5 x}$ $\frac{2}{5}$ $\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 4 x}$ $\frac{1}{2}$ $\lim_{x \to 0} \frac{\sin (- 6 x)}{3 x}$ $- 2$ $\lim_{x \to 0} \frac{e^{6 x} - 1}{5 x}$ $\frac{6}{5}$ $\lim_{x \to 0} \frac{e^{6 x} - 1}{- x}$ $- 6$ $\lim_{x \to 0} \frac{e^{- 6 x} - 1}{6 x}$ $- 1$ $\lim_{x \to 0} \frac{e^{- 5 x} - 1}{x}$ $- 5$ $\lim_{x \to 0} \frac{e^{4 x} - 1}{x}$ $4$ $\lim_{x \to 0} \frac{e^{2 x} - 1}{- 2 x}$ $- 1$ $\lim_{x \to 0} \frac{\sin (- x)}{4 x}$ $- \frac{1}{4}$ $\lim_{x \to 0} \frac{e^{- x} - 1}{- 2 x}$ $\frac{1}{2}$ $\lim_{x \to 0} \frac{\sin (x)}{4 x}$ $\frac{1}{4}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

L'Hopital's RuleAnswer keyVersion 1

Date Period

Evaluate each limit.

$\lim_{x \to 0} \frac{e^{x} - 1}{6 x}$ $\frac{1}{6}$
$\lim_{x \to 0} \frac{e^{- 4 x} - 1}{3 x}$ $- \frac{4}{3}$
$\lim_{x \to 0} \frac{\sin (- 3 x)}{- 3 x}$ $1$
$\lim_{x \to 0} \frac{e^{- 3 x} - 1}{- 3 x}$ $1$
$\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 3 x}$ $\frac{2}{3}$
$\lim_{x \to 0} \frac{e^{2 x} - 1}{- 4 x}$ $- \frac{1}{2}$
$\lim_{x \to 0} \frac{e^{2 x} - 1}{3 x}$ $\frac{2}{3}$
$\lim_{x \to 0} \frac{\sin (2 x)}{x}$ $2$
$\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 5 x}$ $\frac{2}{5}$
$\lim_{x \to 0} \frac{e^{- 2 x} - 1}{- 4 x}$ $\frac{1}{2}$
$\lim_{x \to 0} \frac{\sin (- 6 x)}{3 x}$ $- 2$
$\lim_{x \to 0} \frac{e^{6 x} - 1}{5 x}$ $\frac{6}{5}$
$\lim_{x \to 0} \frac{e^{6 x} - 1}{- x}$ $- 6$
$\lim_{x \to 0} \frac{e^{- 6 x} - 1}{6 x}$ $- 1$
$\lim_{x \to 0} \frac{e^{- 5 x} - 1}{x}$ $- 5$
$\lim_{x \to 0} \frac{e^{4 x} - 1}{x}$ $4$
$\lim_{x \to 0} \frac{e^{2 x} - 1}{- 2 x}$ $- 1$
$\lim_{x \to 0} \frac{\sin (- x)}{4 x}$ $- \frac{1}{4}$
$\lim_{x \to 0} \frac{e^{- x} - 1}{- 2 x}$ $\frac{1}{2}$
$\lim_{x \to 0} \frac{\sin (x)}{4 x}$ $\frac{1}{4}$

How to do these

$\lim_{x \to 0} \frac{\sin (3 x)}{5 x}$

At x equals zero the top and the bottom are both zero, so the rule applies.
Differentiate each separately: the top becomes 3 cosine 3x and the bottom becomes 5.
Now substitute: 3 times 1 over 5, which is three fifths.

The answer is $\frac{3}{5}$ .

Where students go wrong

Using the quotient rule. The rule differentiates the top and the bottom separately and then divides, which is a different thing entirely, and checking that the limit really is 0/0 first matters just as much.

Also called L'Hospital's rule, indeterminate form, zero over zero or infinity over infinity.

Questions about these worksheets

Yes. Every l'hopital's rule 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 l'hopital's rule 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.

L'Hopital's rule worksheet

Select your difficulty

L'Hopital's Rule

L'Hopital's RuleAnswer keyVersion 1

How to do these

Where students go wrong

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