\lim_{\Delta z\to 0}\frac{|z_0+\Delta z|-|z_0|}{\Delta z}
By triangle inequality,
|z_0+\Delta z|-|z_0| \leq |\Delta z|
Using special case where,
|z_0+\Delta z|-|z_0| = |\Delta z|
Gives,
\lim_{\Delta z\to 0}\frac{|\Delta z|}{\Delta z}
When approaching zero on positive real axis, this...