Key takeaways
- Use L’Hôpital’s Rule only for 0/0 or ∞/∞ forms.
- Differentiate the top and bottom separately, then take the limit again.
- Do not use the quotient rule. It is a different thing.
- For (xⁿ − aⁿ)/(x − a) as x → a, the limit is n·aⁿ⁻¹.
In this guide
What is L’Hôpital’s Rule?
A limit asks what value a function gets closer and closer to as x approaches a number. Usually you just plug the number in. But for (x² − 9)/(x − 3) at x = 3, plugging in gives 0/0.
L’Hôpital’s Rule says: when you get 0/0 (or ∞/∞), replace the top and bottom with their derivatives, then try the limit again.
Why does L’Hôpital’s Rule matter?
0/0 limits appear all the time in calculus. The definition of a derivative itself is a 0/0 limit. Without a tool like this, you would need clever algebra or tables of values for each one.
The rule is named after the French Marquis de l’Hôpital, who published it in 1696 in an early calculus textbook. His teacher, Johann Bernoulli, had actually discovered it.
The rule
| Symbol | What it means | Example |
|---|---|---|
| f(x) | The top (numerator) | x² − 9 |
| g(x) | The bottom (denominator) | x − 3 |
| f′, g′ | Their derivatives, taken separately | 2x and 1 |
| a | The number x is approaching | 3 |
Why it works: near x = a, both f and g behave like straight lines through zero, so their ratio is close to the ratio of their slopes.
This calculator works on the limit of (xⁿ − aⁿ)/(x − a) as x → a. The top’s derivative is n·xⁿ⁻¹ and the bottom’s is 1, so the answer is n·aⁿ⁻¹.
How to use L’Hôpital’s Rule step by step
- 1Plug in x = a first. If you get a normal number, that is the answer. Stop.
- 2Check the form. Only continue if you get 0/0 or ∞/∞.
- 3Differentiate top and bottom separately. Find f′(x) and g′(x).
- 4Plug in again. Work out f′(a)/g′(a).
- 5Repeat if needed. If you still get 0/0, differentiate again.
Now try it yourself. Enter n and a in the calculator below to see each step of the rule.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using it when the form is not 0/0 or ∞/∞.
lim (x→0) (x + 1)/(x + 2)is simply 1/2. Using the rule wrongly would give 1/1 = 1. - Forgetting to check again. After one round, plug in before differentiating a second time.
- Using it on other forms directly. Forms like 0 × ∞ must first be rewritten as a fraction.
Tips and tricks
- Famous limits:
lim (x→0) sin x / x = cos 0 / 1 = 1andlim (x→0) (eˣ − 1)/x = e⁰/1 = 1. - Numbers check: try x = 3.001 on a calculator.
(3.001² − 9)/0.001 = 6.001, very close to 6. - Other tools: the squeeze theorem helps with limits that bounce around, like x·sin(1/x).
Where you’ll use it in real life
- Derivatives from scratch: the limit definition calculator is built on 0/0 limits.
- Continuity: filling a “hole” in a graph so the function is continuous (see the continuity check calculator).
- Comparing growth: deciding which of two quantities grows faster, like x² versus eˣ.
- Physics and engineering: simplifying formulas at special values where they seem to break.
Quick summary
When a limit gives 0/0 or ∞/∞, differentiate the top and bottom separately and try again. Always check the form first, and never use the quotient rule here. Use the calculator above for limits of the form (xⁿ − aⁿ)/(x − a), with every step shown.
Worked example
Your homework asks for the limit of (x² − 9)/(x − 3) as x approaches 3. Plugging in gives 0/0, so you use L’Hôpital’s Rule. This has the form (xⁿ − aⁿ)/(x − a) with n = 2 and a = 3.
- 01Limit: lim[x→3] (x^2 − 3^2) / (x − 3)
- 02Direct substitution gives 0/0 → indeterminate form
- 03Apply L'Hôpital's Rule: differentiate numerator and denominator
- 04f(x) = x^2 − 3^2 → f'(x) = 2x^1
- 05g(x) = x − 3 → g'(x) = 1
- 06lim[x→3] f'(x)/g'(x) = 2x^1|_{x=3}
- 07= 2 × 3^1 = 2 × 3 = 6
These numbers are pre-filled in the calculator above.
Frequently asked questions
When can I use L'Hôpital's Rule?
Only when plugging in gives an indeterminate form: 0/0 or ∞/∞. Other forms like 0 × ∞ must first be rewritten as a fraction.
Can I use L'Hôpital's Rule more than once?
Yes. If the new limit is still 0/0 or ∞/∞, differentiate the top and bottom again, as many times as needed.
Is L'Hôpital's Rule the same as the quotient rule?
No. The quotient rule finds the derivative of a fraction. L’Hôpital’s Rule finds a limit by differentiating the top and bottom separately.
