Math
L'hopital's rule.
Definition
A method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. If lim f(x)/g(x) gives an indeterminate form, you can take the derivative of the numerator and denominator separately and re-evaluate the limit.
How it works · 4 phases
Step by step.
- Verify the limit produces an indeterminate form (0/0 or ∞/∞).
- Differentiate the numerator and denominator separately (do not use the quotient rule).
- Evaluate the new limit of f′(x)/g′(x).
- If still indeterminate, apply the rule again.
Examples
Real-world.
- 1 lim as x→0 of sin(x)/x = lim of cos(x)/1 = 1
- 2 lim as x→∞ of ln(x)/x = lim of (1/x)/1 = 0
Key Fact
If lim f(x)/g(x) is 0/0 or ∞/∞, then lim f(x)/g(x) = lim f′(x)/g′(x)