L'Hôpital's Rule can be applied when a limit has which form?

Question

Accepted Answer

0/0 or ∞/∞. The correct answer is 0/0 or ∞/∞. L'Hôpital's Rule states that if lim f(x)/g(x) produces an indeterminate form 0/0 or ∞/∞, then the limit equals lim f'(x)/g'(x), provided this latter limit exists. The rule applies specifically and only to these two indeterminate quotient forms.

Claiming the rule applies to any difficult limit is incorrect; L'Hôpital's Rule has specific prerequisites and cannot be applied to limits that are not indeterminate or not in quotient form. The forms 0·∞ and ∞ - ∞ are indeed indeterminate, but they must first be algebraically rewritten as 0/0 or ∞/∞ before L'Hôpital's Rule can be used. The forms 1^∞ and 0⁰ are also indeterminate but require logarithmic techniques to convert them into applicable forms.

L'Hôpital's Rule is one of the most powerful and frequently tested limit evaluation tools on the CLEP exam. Students must verify the indeterminate form before applying the rule (applying it to a non-indeterminate form gives wrong answers). They should also know how to convert other indeterminate forms (0·∞, ∞-∞, 1^∞, 0⁰, ∞⁰) into 0/0 or ∞/∞ through algebraic manipulation.

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