L'Hôpital's Rule

Evaluating limits of indeterminate forms 0/0 and ∞/∞

Golden Rule: Only apply L'Hôpital when you have confirmed an indeterminate form (0/0 or ∞/∞). Differentiate numerator and denominator separately — do NOT use the quotient rule.

• Step 1: Confirm the indeterminate form — Direct substitution must give 0/0 or ∞/∞. If substitution gives a finite/nonzero result, that is the limit — do not apply L'Hôpital. If you get 0·∞ or ∞−∞, rewrite as a fraction first.
• Step 2: Differentiate top and bottom separately — Differentiate the numerator f(x) to get f'(x), then differentiate the denominator g(x) to get g'(x). The new limit is lim f'(x)/g'(x). This is NOT the quotient rule.
• Step 3: Evaluate the new limit — Substitute x = a (or x → ∞) into f'(x)/g'(x). If it is no longer indeterminate, that is your answer. If it is still 0/0 or ∞/∞, apply L'Hôpital again.