Math Help - Complex Limit

1. Complex Limit

Consider the Limit $\displaystyle \lim_{z\to i}\left[|z|+iArg(iz)\right]$.

What is the limit when approaching i on the unit circle in the first quadrant?

$\displaystyle \lim_{z\to i}|z|+\lim_{z\to i}iArg(iz)\rightarrow 1+\lim_{\theta\to \frac{\pi}{2}}-\theta\rightarrow 1-\frac{\pi}{2}$

Is this correct?

2. No.

$\displaystyle \lim_{z \to i}i\arg{(iz)} = i\arg{(i^2)} = i\arg{(-1)} = \pi i$.