Question 2.30: Prove that limz→0 (z/z) does not exist.
Prove that \lim _{z \rightarrow 0}(\bar{z} / z) does not exist.
If the limit is to exist, it must be independent of the manner in which z approaches the point 0 .
Let z \rightarrow 0 along the x axis. Then y=0, and z=x+i y=x and \bar{z}=x-i y=x, so that the required limit is
Let z \rightarrow 0 along the y axis. Then x=0, and z=x+i y=i y and \bar{z}=x-i y=-i y, so that the required limit is
\lim _{y \rightarrow 0} \frac{-i y}{i y}=-1Since the two approaches do not give the same answer, the limit does not exist.
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