Question

use the definition to the derivative of :
f(z)=(2z-i)/(z-2i) at z=-i .

Answer 1

So, what's the problem here. The definition of derivative here is the limit as z --> i of
\[\frac{f(z) - f(-i)}{z-(-i)}\]

So write down that ratio and start simplifying until you can take the limit.

Answer 2

the problem is that i get stuck half way before i can apply the limit ..

Answer 3

One way to go is define z= -i + delta, and let delta approach zero:
\[\lim_{\Delta \rightarrow 0}\frac{f(-i+\Delta)-f(-i)}{(-i+\Delta)-(-i)}\]
This turns into
\[\lim_{\Delta \rightarrow 0}\frac{1}{\Delta}\left\{ \frac{2(\Delta-i)-i}{(\Delta-3i)} -\left( \frac{-2i-i}{-3i} \right)\right\}\]
Using a common denominator of
\[-3i(\Delta-3i)\]
add the two fractions, divide out terms, and then let Delta go to zero