Definition: L'Hopital's rule evaluates a limit of the form 0/0 or infinity/infinity by replacing the quotient of functions with the quotient of their derivatives.
What does the rule say?
If lim f(x)/g(x) produces 0/0 or ∞/∞, then the limit equals lim f'(x)/g'(x), provided that second limit exists. You differentiate top and bottom separately.
How do you use it?
Worked example: limx→0 (e3x – 1)/sin(5x). Substitution gives (1 – 1)/0 = 0/0, so the rule applies. The derivatives are 3e3x on top and 5 cos(5x) on the bottom. At x = 0 they become 3 and 5, so the limit is 3/5.
A table supports it: at x = 0.001 the quotient is about 0.6005.
What goes wrong?
Using the quotient rule instead of differentiating separately. Another error is applying the rule when the form is not indeterminate, such as 2/0. Mathbot first checks the form, then differentiates numerator and denominator one at a time.
Where to learn it
- Limits and Continuity: Approaching a Value – lesson
- Derivative Definition and Differentiation Rules – lesson
Related terms
L’Hopital’s rule: key points
Before you use L’Hopital’s rule, check that the limit really has the form 0/0 or infinity/infinity; otherwise the rule gives a wrong answer. Then differentiate the numerator and the denominator separately, never with the quotient rule. If the new limit still has an indeterminate form, you may apply L’Hopital’s rule again. For example, the limit of (1 − cos x)/x² as x approaches 0 needs the rule twice and equals 1/2. Forms such as 0 · infinity must first be rewritten as a quotient.