Please don’t confuse these three formulas:

1. Limit of a quotient: lim_{x→a} \frac{f(x)}{g(x)} = \frac{lim_{x→a} f(x)}{lim_{x→a} g(x)} (if the limits exist and that of g is not zero).

2. Derivative of a quotient: \frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f′(x)g(x) − f(x)g′(x)}{g(x)^{2}} .

3. L’Hôpital’s rule: lim_{x→a} \frac{f(x)}{g(x)} = lim_{x→a} \frac{f′(x)}{g′(x)} if the original limit was an indeterminate form of type \frac{0}{0} or \frac{∞}{∞} (and the new limit exists).