L'hopital's rule

Math

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

  1. Verify the limit produces an indeterminate form (0/0 or ∞/∞).
  2. Differentiate the numerator and denominator separately (do not use the quotient rule).
  3. Evaluate the new limit of f′(x)/g′(x).
  4. If still indeterminate, apply the rule again.

Examples

  • lim as x→0 of sin(x)/x = lim of cos(x)/1 = 1
  • 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)

Related Terms

Limit definition Derivative definition Continuity Squeeze theorem

Study This Concept

Practice L'Hopital's rule with free review games in these units:

AP Calculus AB Unit 4: Contextual Applications of Differentiation