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I've read the following claim, and wander how to prove or disprove it:

For a given function $f(z)=g(z)\cdot h(z)$ so that $\lim_{z\to0}g(z)=a$,

there is $h(z)$ so that $f(z)=g(z)\cdot [a+h(z)]$

and $\lim_{z\to0}h(z)=0$

Can someone explain that?

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    My question is wrong.2012-02-17

1 Answers 1

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If I understand you correctly then this statement is false: $\lim_{z\to 0}g(z)=a$ $\lim_{z\to 0}h(z)=0$ Which means $\lim_{z\to 0}f(z)=\lim_{z\to 0}g(z)\lim_{z\to 0}h(z)=0\neq \lim_{z\to 0}f(z)=\lim_{z\to 0}g(z)\lim_{z\to 0}[a+h(z)]=a^2$

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    One more uncertainty: is the question about proving/disproving $f(z)=g(z)h(z)=(a+r(z))h(z)$ or $\lim_{z\to 0}f(z)=\lim_{z\to 0}g(z)h(z)=\lim_{z\to 0}(a+r(z))h(z)$ ?2012-02-17