yet another limit question

Hi,

I have banged my head against this wall for hours:

$\displaystyle lim_{z \to 1-i} \ \ x+i2x + iy = 1 + i$

$\displaystyle

z \in C, z=x+iy

$

I need to prove that the above is true, but without resorting to the theorem that limit of P(z) at a, where z and a are complex numbers, is P(a). nor do I want to use the theorem on limits of products or limit of two functions added together.

I basically want to construct a Delta that would make the following statement true:

$\displaystyle \forall\epsilon>0,\ \exists \delta>0 {, } \forall{z, 0<}\mid{z-(1-i)}\mid{ < }\delta{, }\mid (x + i2x + iy) - (1+i) \mid < \epsilon$

I have tried a few different things:

1. translated $\displaystyle x + i2x + iy$ to $\displaystyle z + i(z + \overline{z})$. I then calculated $\displaystyle 1/(z-(1-i))$, then multiplied that by the final expression that needs to be bound by $\displaystyle \epsilon$ in order to get a factor that could somehow by bound by a constant. This would have allowed me to work forward from the expression involving $\displaystyle \delta$ to the final expression bound by $\displaystyle \epsilon$.

2. Tried to work directly with the x's and y's

So far nothing has worked.

Any help would be much appreciated.