# Complex analysis limit

1. Sep 20, 2010

### Dunkle

1. The problem statement, all variables and given/known data
Show that $$lim_{z \rightarrow 1+2i} [ix - (x+y)] = -3 + i$$.

2. Relevant equations
$$lim_{z \rightarrow z_0} f(z) = w_0$$ if and only if given $$\epsilon > 0$$ there exists a $$\delta > 0$$ such that $$0 < |z-z_0| < \delta \Rightarrow |f(z)-w_0| < \epsilon$$

3. The attempt at a solution
$$f(z) = ix-(x+y), w_0 = -3+i, z = x+iy, z_0 = 1+2i$$

I calculated the following:

$$|z-z_0| = \sqrt{(x-1)^2+(y-2)^2}$$ and

$$|f(z)-w_0| = \sqrt{(3-x-y)^2+(x-1)^2}$$

I need to somehow find a relationship between these, and this is where I'm struggling. Any help would be appreciated!