# L'Hopital's rule

If a limit gives 0/0 or infinity/infinity, differentiate the top and the bottom separately and try the limit again.

Maths (university) · Part 04: Applications of derivatives · Card 7 of 9 · Last updated 2026-10-19

## Analogy

Zooming in on two fractions both heading to zero: what matters is how fast each one gets there.

## Example

**Limit of (e^x - 1)/x at 0**

Substituting gives 0/0.
Differentiate top: e^x.
Differentiate bottom: 1.
Limit of e^x / 1 at 0 is 1.

## In practice

Resolves many limits that algebra cannot easily simplify.

## Practice

### Flashcards

- **When can you use L'Hopital's rule?** When the limit has the form 0/0 or infinity/infinity.

## Questions people ask

### Do I use the quotient rule in L'Hopital's rule?

No. You differentiate the numerator and the denominator separately, then form a new fraction. The quotient rule would be used only if you were finding the derivative of the whole fraction. Mixing these up is a very common mistake. For (e to the x - 1) over x, the new fraction is e to the x over 1.

### When is L'Hopital's rule not allowed?

Only use it when the limit is 0/0 or infinity over infinity. If the form is something like 3/0 or 0/5, the rule does not apply and gives wrong answers. Always check the form first by substituting the limiting value. For example, the limit of (x + 1)/(x + 2) at 0 is simply 1/2, found by substituting.

## Related concepts

- [Critical point](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-1)
- [First derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-2)
- [Second derivative test](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-3)
- [Mean value theorem](https://studyvisually.com/learn/science/maths-university/maths-university-part-4/maths-university-part-4-concept-5)

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