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Proof of L'Hospital's Rule

  1. Feb 27, 2010 #1
    Suppose f(z) is analytic in a region R including the point z0. Prove that f(z) = f(z0) + f'(z0)(z-z0) + η(z-z0) where η ~> 0 as z ~> z0.

    this is actually a lemma my book proves first before actually proving L'Hospital's rule. I understood how they used the lemma to prove the rule but i don't really understand the logic in proving this lemma. my book did:

    Let [f(z) - f(z0)]/(z-z0) - f'(z0) = η so that f(z) = f(z0) + f'(z0)(z - z0) = η(z-z0).
    Then, since f(z) is analytic at z0, we have as required:
    lim (z ~> z0) of η = lim (z ~> z0) of [f(z) - f(z0)]/(z-z0) - f'(z0) = f'(z0) - f'(z0) = 0.

    i don't understand how f(z) = f(z0) + f'(z0)(z - z0) = η(z-z0). shouldn't it be f(z) = η(z-z0) + f'(z0)(z - z0) + f(z0) since they let [f(z) - f(z0)]/(z-z0) - f'(z0) = η?
     
  2. jcsd
  3. Feb 27, 2010 #2

    tiny-tim

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    Hi demonelite123! :smile:

    It's just a typo :rolleyes: … + and = are the same key on most keyboards! :wink:
     
  4. Mar 1, 2010 #3
    oh no wonder. thanks!
     
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