L'Hôpital's rule for 0/0 forms states that, if lim f(x)=lim g(x)=0 or both → ∞ as x→a, and f', g' exist near a with g'(x) ≠ 0, then lim_{x->a} f(x)/g(x) equals:

L'Hôpital's rule for 0/0 forms states that, if lim f(x)=lim g(x)=0 or both → ∞ as x→a, and f', g' exist near a with g'(x) ≠ 0, then lim_{x->a} f(x)/g(x) equals:

• A. lim f'(x)/g'(x) (provided it exists)
• B. lim f(x)/g(x)
• C. lim f''(x)/g''(x)
• D. lim f(x) - g(x)

Answer: (A)

L'Hôpital's rule lets you replace a limit of a ratio that forms an indeterminate 0/0 (or ∞/∞) with the limit of the ratio of the derivatives, when the right conditions are met. Here, as x approaches a, both f(x) and g(x) go to zero and the derivatives f' and g' exist near a with g' not zero. If the limit of f'(x)/g'(x) exists (finite or infinite), then the limit of f(x)/g(x) exists and equals that same limit. Intuitively, you can use the Mean Value Theorem: for x near a, there is a c between a and x with f(x) − f(a) over g(x) − g(a) equal to f'(c)/g'(c). If f(a) = g(a) = 0, this becomes f(x)/g(x) = f'(c)/g'(c). As x → a, c → a, so the limit of f(x)/g(x) mirrors the limit of f'(c)/g'(c), which tends to the limit of f'(x)/g'(x) if that limit exists. That’s why the correct expression is the limit of f'(x)/g'(x). If that derivative ratio doesn’t have a (finite or infinite) limit, L'Hôpital doesn’t give a conclusion about f(x)/g(x).

L'Hôpital's rule lets you replace a limit of a ratio that forms an indeterminate 0/0 (or ∞/∞) with the limit of the ratio of the derivatives, when the right conditions are met. Here, as x approaches a, both f(x) and g(x) go to zero and the derivatives f' and g' exist near a with g' not zero. If the limit of f'(x)/g'(x) exists (finite or infinite), then the limit of f(x)/g(x) exists and equals that same limit.

Intuitively, you can use the Mean Value Theorem: for x near a, there is a c between a and x with f(x) − f(a) over g(x) − g(a) equal to f'(c)/g'(c). If f(a) = g(a) = 0, this becomes f(x)/g(x) = f'(c)/g'(c). As x → a, c → a, so the limit of f(x)/g(x) mirrors the limit of f'(c)/g'(c), which tends to the limit of f'(x)/g'(x) if that limit exists. That’s why the correct expression is the limit of f'(x)/g'(x). If that derivative ratio doesn’t have a (finite or infinite) limit, L'Hôpital doesn’t give a conclusion about f(x)/g(x).