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L'Hopital's Rule doubt

  1. Oct 12, 2004 #1
    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. jcsd
  3. Oct 12, 2004 #2
    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
  4. Oct 12, 2004 #3


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    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.
  5. Oct 12, 2004 #4


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    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 [tex]f(x)\approx{f}(a)+f'(a)(x-a)[/tex]
    in a neighbourhood of x=a. (This is a first-order Taylor-approx.)
    Similarly, we have a function [tex]g(x)\approx{g}(a)+g'(a)(x-a)[/tex]
    Hence we have that [tex]h(x)=\frac{f(x)}{g(x)}\approx\frac{f(a)+f'(a)(x-a)}{g(a)+g'(a)(x-a)}[/tex]
    in the same neighbourhood.
    We are interested in [tex]\lim_{x\to{a}}h(x)[/tex]
    Furthermore, we assume f(a)=g(a)=0, that is:
    [tex]h(x)\approx\frac{f'(a)}{g'(a)}[/tex] 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
  6. Oct 13, 2004 #5
    Thank you all of you for your contribution. Finally, I got it.
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