I have two statements, and I am working in the complex plane:
(1): $\underset{w\rightarrow z}{lim}g(w)=0$
(2): $\underset{w\rightarrow z}{lim}\frac{g(w)}{w-z}=0$
I want to know which one implies the other. I am pretty sure that (2)->(1), however I am having difficulties proving it.
I started by assuming (2). I then rewrite (2) using the limit law for fractions and get:
$\frac{\underset{w\rightarrow z}{lim}g(w)}{\underset{w\rightarrow z}{lim}w-z}=0$.
Now, I know that $\underset{w\rightarrow z}{lim}(w-z)=0$ but now can i conclude anything:
$\frac{\underset{w\rightarrow z}{lim}g(w)}{0}=0$ ?
I was thinking going the contrapositive way and assuming that (1) is false. This would imply that $\underset{w\rightarrow z}{lim}g(w)\neq0$ and therefore $\frac{\underset{w\rightarrow z}{lim}g(w)}{\underset{w\rightarrow z}{lim}w-z}=\infty$.
any suggestions ?
Also,a counterexample of some function g that satisfies (1) and not (2) would be most helpful...