L'Hôpital's rule applies to which forms, provided the standard conditions hold?

• A. 0/0 or ∞/∞ forms
• B. Any ratio with a finite limit
• C. Only 1/0 forms
• D. Only when the numerator is zero everywhere

Answer: (A)

L'Hôpital's rule is used when a limit looks indeterminate because the numerator and denominator both approach zero or both grow without bound as the variable approaches a point. Under the standard conditions—both functions are differentiable near that point, the derivative of the denominator isn’t zero near there, and the limit of the ratio of the derivatives exists (finite or infinite)—the original limit equals the limit of the ratio of the derivatives. This is why the applicable forms are 0/0 and ∞/∞: in those cases the rule lets you move from an indeterminate original expression to a determinate limit of the derivatives. If the original limit were in other forms (like a finite number over zero, or zero over a finite number) L’Hôpital’s rule doesn’t apply.