Question 2.6.1: A Limit That Does Not Exist. Show that limz→0 z/z does not e......
A Limit That Does Not Exist
Show that \underset{z\rightarrow 0}{lim} \frac{z}{\overline{z} } does not exist.
We show that this limit does not exist by finding two different ways of letting z approach 0 that yield different values for \underset{z\rightarrow 0}{lim} \frac{z}{\overline{z} }. First, we let z approach 0 along the real axis. That is, we consider complex numbers of the form z = x + 0i where the real number x is approaching 0. For these points we have:
\underset{z\rightarrow 0}{lim} \frac{z}{\overline{z} }=\underset{x\rightarrow 0}{lim} \frac{x + 0i}{x – 0i} =\underset{x\rightarrow 0} {lim} 1 = 1 (2)
On the other hand, if we let z approach 0 along the imaginary axis, then z = 0+iy where the real number y is approaching 0. For this approach we have:
\underset{z\rightarrow 0}{lim} \frac{z}{\overline{z} }=\underset{y\rightarrow 0}{lim}\frac{0+yi}{0-yi} =\underset{y\rightarrow 0} {lim}(-1)=-1 (3)
Since the values in (2) and (3) are not the same, we conclude that \underset{z\rightarrow 0}{lim} \frac{z}{\overline{z} }
does not exist.