Suppose f(z) is analytic in a region R including the point z(adsbygoogle = window.adsbygoogle || []).push({}); _{0}. Prove that f(z) = f(z_{0}) + f'(z_{0})(z-z_{0}) + η(z-z_{0}) where η ~> 0 as z ~> z_{0}.

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(z_{0})]/(z-z_{0}) - f'(z_{0}) = η so that f(z) = f(z_{0}) + f'(z_{0})(z - z_{0}) = η(z-z_{0}).

Then, since f(z) is analytic at z_{0}, we have as required:

lim (z ~> z_{0}) of η = lim (z ~> z_{0}) of [f(z) - f(z_{0})]/(z-z_{0}) - f'(z_{0}) = f'(z_{0}) - f'(z_{0}) = 0.

i don't understand how f(z) = f(z_{0}) + f'(z_{0})(z - z_{0}) = η(z-z_{0}). shouldn't it be f(z) = η(z-z_{0}) + f'(z_{0})(z - z_{0}) + f(z_{0}) since they let [f(z) - f(z_{0})]/(z-z_{0}) - f'(z_{0}) = η?

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# Proof of L'Hospital's Rule

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