When Can i apply L'Hopital's rule?

[zolit]

I am trying to work through the following problem:
if function is differentiable on an interval containing 0 except possibly at 0, and it is continuous at 0, and 0= f(0)= lim f ' (x) (as x approaches 0). Prove f'(0) exists and = 0.

I thought of using the definition of a limit to get to lim [ f(x)/x] then set g(x)=x and then use L'hopital's rule. The problem is - I'm not sure I can, is it enough to show that both f, g go to zero as x goes to zero to use it??

The alternative approach i was thinking of is considering two intervals (minus delta, 0) and (0, plus delta) and then using Rolle's theorem, but the solution seems to get too complicated from then.

 

[copperboy]

zolit said -- "I thought of using the definition of a limit to get to lim [ f(x)/x] then set g(x)=x and then use L'hopital's rule."

My point of view is this, when it comes to prove something, you can either based your solutions on definition or some existing theorem. L'hopital's rule requires that both f'(x) and g'(x) exist. And here in your problem you are asked to prove that f'(x) exists when x=0.

The approach I was thinking of is considering f'+(0) and f'-(0) both exist and are equal to each other.

 

[M98Ranger]

L'Hopital's rule can be applied when we have an indeterminate form, such as 0/0 or infinity/infinity. In this case, it seems like you have correctly identified that the use of L'Hopital's rule is possible, as the limit you are trying to evaluate is in the form of 0/0.

To apply L'Hopital's rule, we need to have a differentiable function in both the numerator and denominator of the fraction. In this case, since f is differentiable on the interval containing 0, we can use L'Hopital's rule to evaluate the limit.

However, it is important to note that L'Hopital's rule is not always the best approach to solving a limit problem. In some cases, as you have mentioned, using other techniques such as Rolle's theorem may lead to a simpler solution. Ultimately, it is up to you as the problem solver to decide which method is most appropriate in each situation.

In this particular problem, it seems like using L'Hopital's rule is a valid approach. So, you can go ahead and use it to prove that f'(0) exists and is equal to 0. Just make sure to carefully check the conditions for applying L'Hopital's rule and to show all the necessary steps in your solution. Good luck!