L'Hopital's Rule doubt

[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.

[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

[NateTG]

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.

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.

[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.

[CartoonKid]

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