# L'Hopital's Rule doubt

1. Oct 12, 2004

### CartoonKid

Can anyone tell me, what is the rationale behind L'Hopital's Rule? I just know that how to use it but don't know why it is logic.

2. Oct 12, 2004

### TenaliRaman

its not any logic ... it has good enough proof ....
try proving it if u want ... shouldn't be hard ....
it should give u a good rundown on all the properties of limits :)

-- AI

3. Oct 12, 2004

### NateTG

It's possible to prove the various versions of L'Hopital's rule using epsilons and deltas. It's really, more or less, a shortcut like all of the derivative and integral formulae.

4. Oct 12, 2004

### arildno

One easy way to think of L'Hopital's rule in its most usual form, is to consider the Taylor series approximations of the two functions:
Suppose a function $$f(x)\approx{f}(a)+f'(a)(x-a)$$
in a neighbourhood of x=a. (This is a first-order Taylor-approx.)
Similarly, we have a function $$g(x)\approx{g}(a)+g'(a)(x-a)$$
Hence we have that $$h(x)=\frac{f(x)}{g(x)}\approx\frac{f(a)+f'(a)(x-a)}{g(a)+g'(a)(x-a)}$$
in the same neighbourhood.
We are interested in $$\lim_{x\to{a}}h(x)$$
Furthermore, we assume f(a)=g(a)=0, that is:
$$h(x)\approx\frac{f'(a)}{g'(a)}$$ close enough.
L'Hopital's rule states that this is, in fact the limit of h(x) as x goes to a.

Last edited: Oct 12, 2004
5. Oct 13, 2004

### CartoonKid

Thank you all of you for your contribution. Finally, I got it.