f(z) limit using formal definition

Meggle

1. The problem statement, all variables and given/known data
For each of the following choices of f(z) use the definition of a limit to obtain lim z-->0 f(z) or prove that the limit doesn't exist
(a) [tex]\frac{|z|^{2}}{z}[/tex]


2. Relevant equations
Formal limit definition


3. The attempt at a solution
f(z) = [tex]\frac{|z|^{2}}{z}[/tex]
f(z) = [tex]\frac{x^{2} + y^{2}}{x +iy}[/tex]
So if z=(x,0), f(z)=[tex]\frac{x^{2} + 0}{x +i0}[/tex] = x
Then f(z) --> 0 as (x,y) --> 0 along the real axis
And if z=(0,y), f(z)=[tex]\frac{0 + y^{2}}{0 +iy}[/tex] = [tex]\frac{y}{i}[/tex]
Then f(z) --> 0 as (x,y) --> 0 along the imaginary axis
So maybe lim z-->0 f(z)= 0
Suppose lim z-->0 f(z)= 0, then for each [tex]\epsilon[/tex] >0 there exists [tex]\delta[/tex] >0 such that 0 < |z - 0| < [tex]\delta[/tex] implies |f(z) - 0| < [tex]\epsilon[/tex]
.....
Sooooo how do I figure delta out of that? I can't see how to simplify it or what to do next. It's due tomorrow, of course. And there's a (b) and a (c) , but think I could work them out if I could finish this one.

Also, can anyone tell me how to make the formulas update? I've changed all the SUP to curly brackets and carrots, but it wont seem to referesh. Edit never mind, seems they just don't show right on preview.

CompuChip

Here's a suggestion:
[tex]\left| \frac{z}{|z|} \right| = 1 [/tex]
So if you write
[tex] |f(z)| = \left| |z| \cdot \left( \frac{z}{|z|} \right)^{-1} \right| [/tex]
you can work in your delta.

╔(σ_σ)╝

Epsilon= delta. Try to justify it.