A simple question : Is $g(z + \Delta z).g(z) = g(z)^2$? Is $g(z + \Delta z).g(z) = g(z)^2$ ?
The full expression is a $\lim_{\Delta z \to 0} \frac {A}{G} $, where $G$ is the left-hand side of the above expression. 
It's a question about a solution to a problem that I'm going through, and I can't post it because I don't have enough reps. But the gist of it is that the above expression is taken to be true and thus he factors a $g(z)^2$ out of the limits in the denominator. 
I'd appreciate it if somebody explains how this is possible. Thanks! 

Let us see a general answer to your particular question. Let $g$ be a function, and assume that $g$ is continuous at $z_0$. Then
$$
\lim_{\Delta z \to 0} g(z_0+\Delta z) = g(z_0), \tag{1}
$$
and hence
$$
\lim_{\Delta z \to 0} g(z_0+\Delta z)g(z_0) = g(z_0)\cdot g(z_0)=g(z_0)^2.
$$
Since (1) is the definition of continuity, the answer to your questoin is affirmative provided that $g$ is continuous. If $g$ is not supposed to be continuous, then then answer might be negative: you should be able to use any reasonable kind of discontinuous counterexample to check this.
