(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Use epsilon-delta proof to show that [tex]\lim_{z\to z_0}(z^2+c)=z_0^2+c[/tex].

2. Relevant equations

[tex]\forall\epsilon>0 \exists\delta>0 \forall z (|z-z_0|<\delta\Rightarrow|f(z)-\omega_0|<\epsilon)[/tex]

3. The attempt at a solution

So [tex]f(z)=z^2+c[/tex] and [tex]\omega_0=z_0^2+c[/tex]. In order to write my proof I first need to find delta in terms of epsilon.

Let [tex]\epsilon>0[/tex]. Then

[tex]\begin{align*}|f(z)-\omega_0|<\epsilon &\Rightarrow|(z^2+c)-(z_0^2+c)|<\epsilon \\

&\Rightarrow|z^2-z_0^2|<\epsilon\\

&\Rightarrow|(z+z_0)(z-z_0)|<\epsilon\\

&\Rightarrow|z+z_0||z-z_0|<\epsilon\\

&\Rightarrow|z-z_0|<\dfrac{\epsilon}{|z+z_0|}\end{align*}[/tex]

And from here I'm kind of stumped. I need to get that z out of the right hand side so that i can choose delta in terms of epsilon only. I know that with functions of a real variable you can restrict the value of [tex]|x-x_0|<1[/tex] or something like that so you can put an upper and lower bound on delta, but I am not sure exactly how to employ this technique since pulling z's out of absolute value sign is a little different that pulling out x's.

Any help would be greatly appreciated. Thanks in advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Limit of a Complex Function

**Physics Forums | Science Articles, Homework Help, Discussion**