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

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Homework Statement



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

Homework Equations





The Attempt at a Solution


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Write it out with z=x+iy.
 
Take the limit approaching 0 along the real axis and along the imaginary axis. Show that the results are different.
 
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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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