Which statement best expresses a condition for applying L'Hôpital's Rule to lim_{x->a} f(x)/g(x)?

Which statement best expresses a condition for applying L'Hôpital's Rule to lim_{x->a} f(x)/g(x)?

• A. The limits form 0/0 or ∞/∞ and the derivatives f'(x) and g'(x) have a finite limit as x approaches a.
• B. The limits form are 0/0 or ∞/∞ and the derivatives f'(x) and g'(x) have a finite limit as x approaches a.
• C. The functions are polynomials.
• D. The quotient is defined for all x near a.

Answer: (B)

The main idea is that L’Hôpital’s Rule applies when evaluating a limit that produces an indeterminate form, specifically 0/0 or ∞/∞, and you can relate that limit to the behavior of the derivatives. If as x approaches a, the ratio f(x)/g(x) is in one of those indeterminate forms and f and g are differentiable near a with g′(x) not vanishing there, then the limit equals the limit of f′(x)/g′(x) (provided that latter limit exists and is finite). So the statement that the original limit is in 0/0 or ∞/∞ and that the derivatives have a finite limit as x approaches a matches the situation where using L’Hôpital’s Rule makes sense. Why the other ideas don’t fit as the main condition: being polynomials is not required by L’Hôpital’s Rule, and a quotient being defined near a isn’t enough to justify applying the rule unless the form is indeterminate and the derivative ratio behaves properly.

The main idea is that L’Hôpital’s Rule applies when evaluating a limit that produces an indeterminate form, specifically 0/0 or ∞/∞, and you can relate that limit to the behavior of the derivatives. If as x approaches a, the ratio f(x)/g(x) is in one of those indeterminate forms and f and g are differentiable near a with g′(x) not vanishing there, then the limit equals the limit of f′(x)/g′(x) (provided that latter limit exists and is finite). So the statement that the original limit is in 0/0 or ∞/∞ and that the derivatives have a finite limit as x approaches a matches the situation where using L’Hôpital’s Rule makes sense.

Why the other ideas don’t fit as the main condition: being polynomials is not required by L’Hôpital’s Rule, and a quotient being defined near a isn’t enough to justify applying the rule unless the form is indeterminate and the derivative ratio behaves properly.