Repeated L'Hôpital's Rule

Calculus 1 · Axiom Academy

EXAMPLE Repeated L'Hôpital's Rule When one pass of L'Hôpital still leaves an indeterminate form, differentiate again — and again — until the limit is determinate. Evaluate the limit . Direct substitution gives the indeterminate form , so L'Hôpital's rule applies — but watch what happens after the first pass. Nicely done — you applied L'Hôpital's rule twice , re-checking the form after each pass instead of stopping early. Re-check every pass: L'Hôpital's rule only applies to or . After differentiating, substitute again and confirm the new form before deciding whether you are done. Repeat as needed: was still , so a second differentiation was required to reach the determinate . Whenever a limit stays indeterminate, you can keep differentiating numerator and denominator — the rule is happy to run as many passes as the problem needs.

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