L’Hôpital’s rule
Evaluate certain indeterminate quotients by differentiating numerator and denominator separately.
L’Hôpital’s rule is one of 2 limits formulas in the calculus section of this library, and it is used at ap · university level.
Why l’hôpital’s rule works
Near the trouble spot both the top and the bottom are close to zero, so each behaves like its own tangent line: a slope times the distance from that spot. The shared distance factor cancels between them and leaves the ratio of the two slopes, which is why comparing derivatives answers the question.
What each symbol means
$f,g$ are differentiable near the limit point.
L’Hôpital’s rule: when it holds
The original quotient must have form $0/0$ or $\infty/\infty$, and the derivative quotient limit must satisfy the theorem.
When it stops applying
It applies only to the two indeterminate quotient forms. Used on something like (x+1)/(x+2) as x approaches 0 it returns 1, when the fraction plainly equals one half. Forms such as zero times infinity must be rewritten as a quotient before the rule is allowed.
L’Hôpital’s rule: a worked example
$\lim_{x\to0}\sin x/x=\lim_{x\to0}\cos x/1=1$.
The mistake to avoid
What people do: Applying the quotient rule to the fraction instead of differentiating top and bottom separately.
Why it goes wrong: The rule compares two independent derivatives, while the quotient rule builds a completely different expression, so the limit that comes out is not the one you wanted.
Do this instead: Differentiate the numerator by itself, differentiate the denominator by itself, and take the limit of the new fraction.
L’Hôpital’s rule: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $f,g$ are differentiable near the limit point.
- Check the conditions before substituting. The original quotient must have form $0/0$ or $\infty/\infty$, and the derivative quotient limit must satisfy the theorem.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Limits
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where l’hôpital’s rule comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Understanding Limits — the lesson behind this formula: describe the value a function approaches near a point.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about l’hôpital’s rule
Can I use it more than once?
Yes, as long as each new fraction is still one of the two allowed indeterminate forms. Check the form again before every repeat.
What if the new limit does not exist?
Then the rule tells you nothing either way. The original limit may still exist and has to be found by some other method.
Is it circular to use it on sin x over x?
Slightly, since the derivative of sine is itself proved using that very limit. The squeeze theorem is the honest route to that particular answer.
Does it work at infinity and for one-sided limits?
Yes. The same statement holds as the input approaches a value from one side only, and also as the input grows without bound.