- #1
eptheta
- 64
- 0
Hi,
I am trying to prove that a limit exists at a point using the epsilon delta definition in the complex plane, but I can't seem to reach a conclusion.
Here's what I have been trying to get at:
\lim_{z\to z_o} z^2+c = {z_o}^2 +c
|z^2+c-{z_o}^2-c|<\epsilon \ whenever\ 0<|z-z_o|<\delta
LH=|z^2-{z_o}^2|=|z-z_o||z+z_o|
=|z-z_o||\overline{z+z_o}|
=|z-z_o||\bar{z}+\bar{z_o}|
=|z\bar{z} +z\bar{z_o} -{z_o}\bar{z} -z_o\bar{z_o}|
=| |z|^2 -|z_o|^2 +2Im(zz_o) |
\leq||z|^2 -|z_o|^2 +2|z||z_o|| \ (because\ Im(z)\leq|z|)
But I can't get any further. I did this much thinking I could factor it to the square of delta, but that didn't work out because of the positive 2zzo term.If anyone can help me out here, it would be great. Thanks.
I am trying to prove that a limit exists at a point using the epsilon delta definition in the complex plane, but I can't seem to reach a conclusion.
Here's what I have been trying to get at:
\lim_{z\to z_o} z^2+c = {z_o}^2 +c
|z^2+c-{z_o}^2-c|<\epsilon \ whenever\ 0<|z-z_o|<\delta
LH=|z^2-{z_o}^2|=|z-z_o||z+z_o|
=|z-z_o||\overline{z+z_o}|
=|z-z_o||\bar{z}+\bar{z_o}|
=|z\bar{z} +z\bar{z_o} -{z_o}\bar{z} -z_o\bar{z_o}|
=| |z|^2 -|z_o|^2 +2Im(zz_o) |
\leq||z|^2 -|z_o|^2 +2|z||z_o|| \ (because\ Im(z)\leq|z|)
But I can't get any further. I did this much thinking I could factor it to the square of delta, but that didn't work out because of the positive 2zzo term.If anyone can help me out here, it would be great. Thanks.
