Epsilon-Delta Definition to prove the L'Hopital's Rule

[prasannaworld]

Okay I wish to try to construct an Epsilon-Delta Definition to prove the L'Hopital's Rule (0/0 form). Please correct me if I am wrong.

http://mathforum.org/library/drmath/view/53340.html

I found the above site. Scrolling down one would the proof.

I can follow how an x constraint is constructed. But then for the y constraint, I cannot seem to completely follow the proof when it says:

lim f'(x0)/g'(x0) = L
x0->b-

Any help/guidance will be appreciated

 

[robert Ihnot]

For a continuous function $$limF(x)_{x\rightarrow a}=F(a)$$. In the use of L'Hopital's Rule, we can assume continuity because we are able to employ derivatives.

So that, in general for continuous functions, we can see that $$lim\frac{F(x)}{G(x)}_{x\rightarrow a}=\frac{F(a)}{G(a)}$$ However, in the case of 0/0, we have an underfined quality and need to go further with it, as shown in your reference.