L'Hôpital's rule is used to evaluate limits that yield which indeterminate forms?

L'Hôpital's rule is used to evaluate limits that yield which indeterminate forms?

• A. 0/0 and ∞/∞
• B. 1/0 and ∞/0
• C. 0·∞ and ∞ − ∞
• D. 0/∞ and ∞/0

Answer: (A)

L'Hôpital's rule is used when evaluating limits that form 0/0 or ∞/∞. When you plug in the limit point and both the numerator and denominator approach zero or both grow without bound, the ratio isn’t determined yet. If the functions are differentiable near that point and the derivative of the denominator isn’t zero, you can replace the original limit with the limit of the ratio of their derivatives, often simplifying the evaluation. Other forms like 1/0 or ∞/0 aren’t indeterminate in the same sense—they indicate divergence or undefined behavior, so L'Hôpital's rule doesn’t apply directly. Forms such as 0·∞ or ∞−∞ are indeterminate too, but you typically rewrite them as a quotient first so the 0/0 or ∞/∞ form is achieved before applying the rule.

L'Hôpital's rule is used when evaluating limits that form 0/0 or ∞/∞. When you plug in the limit point and both the numerator and denominator approach zero or both grow without bound, the ratio isn’t determined yet. If the functions are differentiable near that point and the derivative of the denominator isn’t zero, you can replace the original limit with the limit of the ratio of their derivatives, often simplifying the evaluation.

Other forms like 1/0 or ∞/0 aren’t indeterminate in the same sense—they indicate divergence or undefined behavior, so L'Hôpital's rule doesn’t apply directly. Forms such as 0·∞ or ∞−∞ are indeterminate too, but you typically rewrite them as a quotient first so the 0/0 or ∞/∞ form is achieved before applying the rule.