Proving the Limit of (z/\bar{z})2 Does Not Exist

[Rubik]

Homework Statement

Show that the lim z→0 of (z/$\bar{z}$)² does not exist

Homework Equations

The Attempt at a Solution

Not to sure how to go about this question?

 

[micromass]

Write it out with z=x+iy.

 

[HallsofIvy]

Take the limit approaching 0 along the real axis and along the imaginary axis. Show that the results are different.

 

[Rubik]

I have come up with this

Taking the limit along the Real axis:
lim as z→0 of (z/$\bar{z}$)²
= lim (x + 0i)²/(x - 0i)²
= lim x²/x²
= 1

Then taking the limit at the points x + xi for x→0:
lim as z→0 of (z/$\bar{z}$)²
= lim (x + xi)²/(x - xi)²
= lim (2x²)/(-2x²)
= -1

and since 1 ≠ -1 The limit does not exist.