Question

Let f and g be functions that are differentiable for all real numbers, with g(x) ≠0 for x≠0.

Answer 1

If \[\lim_{x \rightarrow 0} f(x)=\lim_{x \rightarrow 0} g(x) \] and \[\lim_{x \rightarrow 0} f \prime(x)/ g \prime(x) \] exists, then
\[\lim_{x \rightarrow 0} f(x)/g(x)\] is

Answer 2

...then
\[ \lim_{x \rightarrow 0} \frac{f(x)}{g(x)} = \lim_{x \rightarrow 0} \frac{f'(x)}{g'(x)} \]

This is called l'Hopitals rule and you've written down all of the conditions necessary to use it. It turns out to be very useful.

Answer 3

that makes sense, thank you!