Teacher page: l'Hopital's rule (AA 5.13)

The one thing to do with the figure

Zoom in and let them see both curves straighten.

Near the point both functions are nearly their own tangents, both passing through zero. Two straight lines through the same point have a ratio equal to the ratio of their gradients, and that is the rule.

Students who see this stop treating it as a conjuring trick, and crucially they stop applying it to limits that are not indeterminate.

Check the form, every single time

The rule applies only to 0/0 and ∞/∞. On anything else it gives a wrong answer with total confidence.

Writing "this is 0/0, so by l'Hopital" takes one line and is usually a mark. Make it compulsory from the first example.

The answers

sin x / x at 0	cos x / 1 = 1.
(e^x − 1)/x at 0	e^x/1 = 1.
(x² − 4)/(x − 2) at 2	2x/1 = 4. Factorising is quicker.
(1 − cos x)/x² at 0	Twice: sin x/2x, then cos x/2 = ½.
1, 2, 3	1; B, just substitute, 3/2; 2 applications.

Where the marks go

1 markStating the indeterminate form before applying the rule.

1 markDifferentiating top and bottom separately and correctly.

1 markRe-checking the form between repeated applications and stopping at the right point.

What each wrong answer tells you

They give	What it means
Using the quotient rule	A method error, not arithmetic. The two look similar on the page and are completely different.
Applying it to 3/2 (Q2)	The target error. Gives 1, confidently, and wrong.
Stopping after one application on Q3	sin x / 2x is still 0/0. Re-check every time.
Continuing past the answer	Differentiating cos x / 2 again is harmless but shows they are not checking.

Other things they will say

"Is factorising allowed instead?" Yes, and for (x²−4)/(x−2) it is quicker. Use whichever is faster; both earn full marks.

"What about 0 times infinity?" Rearrange it into a fraction first, then it is one of the two allowed forms. Worth showing once.

"Why is it called indeterminate?" Because 0/0 can be anything at all, depending on how fast each part goes to zero. That is precisely what the derivatives measure.

A possible order

	What is happening
1	Try to evaluate sin x / x at zero and fail. Establish the problem before the tool.
2	The zoom animation, then the rule stated.
3	Three examples, saying the form out loud each time.
4	Repeated application on (1 − cos x)/x².
5	Questions 1 to 3, with question 2 as the discussion.
6	A set deliberately mixing indeterminate and ordinary limits.

Two things not to say

Do not set an exercise where every limit is indeterminate. Checking the form is the skill, and a page where the answer is always yes destroys it.

Do not call it "differentiating the fraction". That phrasing produces the quotient rule error directly.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

A standard limitEvaluate the limit of (sin x)/x as x tends to 0.
Differentiating top and bottom gives (cos x)/1, which is 1 at x = 0.
AnotherEvaluate the limit of (e^x − 1)/x as x tends to 0.
e^x/1 = 1.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

TwiceEvaluate the limit of (1 − cos x)/x² as x tends to 0.
First pass gives (sin x)/(2x), still 0/0, so apply it again: (cos x)/2 = 0.5.
Check it without the ruleEvaluate the limit of (x² − 4)/(x − 2) as x tends to 2 both ways.
Factorising gives x + 2, which is 4. The rule gives 2x/1 = 4. Factorising first is quicker when it is available.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Check the form firstA student applies the rule to (x + 1)/(x + 2) at x = 0 and gets 1. Explain the error.
The form is 1/2, not 0/0 or infinity over infinity, so the rule does not apply and the answer is simply 1/2. The rule is only valid for an indeterminate form, and checking the form is the first step.
Why the zoom worksExplain geometrically why differentiating top and bottom resolves a 0/0 limit.
Near the point both curves pass through zero, and zooming in makes each indistinguishable from its own tangent. The ratio of the functions becomes the ratio of the tangent gradients, which is the ratio of the derivatives.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.