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