# Does L'Hospital's rule say to take the derivative of the limit or the derivative of the top over the derivative of the bottom?

When using L'Hospital's rule, do you take the derivative of the problem? Or do you put the derivative of the denominator over the derivative of the numerator. For example: does $$\frac{\ln(9-4x)}{\tan(x-2)}$$ become $$\frac{\frac{-4}{9-4x}}{\sec^2(x-2)}$$ or do you need to use the quotient rule?

• No quotient rule Nov 24, 2019 at 20:57
• $\lim \frac{u(x)}{v(x)} = \lim \frac{u'(x)}{v'(x)}$ is what you do Nov 24, 2019 at 21:01

From @GeorgeDewhirst: $$\lim \frac{u(x)}{v(x)} = \lim \frac{u'(x)}{v'(x)}$$ is what you do.
Others have already given the answer - $$\lim \frac{u(x)}{v(x)} = \lim \frac{u'(x)}{v'(x)}$$ is correct - but you should think more about why.
Remember what a limit is - what the value of a function approaches. If the function approaches an indeterminate form, such as $$\frac{0}{0}$$, then it is useful to consider the relative rate of change of numerator and denominator, since this difference is important in evaluating forms such as $$\frac{0}{0}$$. And instantaneous rate of change is just the derivative.