A Limit that Does Not Exist

Consider the example of a potential limit below. $$\lim_{z \rightarrow 0} \frac{ \ z \ }{ \overline{z}}$$

Let's think about what this function, $f(z) = \frac{z}{\overline{z}}$, does to points in the plane goemetrically.

The modulus of $z$ shows up once on the top and is cancelled by the same modulus for $\overline{z}$ on the bottom. The argument ends up doubling, because $\overline{z}$ reverses the argument but dividing reverses it yet again. Expressing $z$ as $z=Re^{i\theta}$, we have $f(z)=e^{i(2\theta)}$.

Drag the point $z$ around in the diagram below. Note that no matter how big or small $z$ is, it still never leaves the unit circle.

Figure 1: $f(z)$

Now, I'm going to make a slight change to the function in order to separate the inputs around the origin from the outputs. Here we modify $f$ to $f(z) = 2 + \frac{ \ z \ }{\overline{z}}$. This just moves all the outputs over by 2.

Take a moment to notice that the real axis has a constant value of 3 and the imaginary axis has a constant value of 1 (except for their intersection, of course).

Figure 2: New $f(z)$

Next, observe that if we let $\delta$ take any positive value, no matter how small, we can pick the points $\frac{\delta}{2}$ and $i\frac{\delta}{2}$ inside the delta ball. Their images will be $3$ and $1$ respectively, which do not change and are two apart from each other.

Figure 3: Fixed images independent of $\delta$

The fact that these images are anchored away from each other makes it impossible for a limit to exist.

In this last figure, $\varepsilon$ is set to be greater than 1 to start. Any potential limit must lie inside both $\varepsilon$ balls because both of those outputs have to be withint $\varepsilon$ of a potential limit.

Drag $\varepsilon$ to a value at or below 1 and see that no limit could exist because there is no overlap. This might be fixed by a smaller $\delta$ in another scenario, but here the images are there for any $\delta$. You can drag the $\delta$ slider to see that making $\delta$ smaller makes no change to this scenario.

Figure 4: Limit DNE

Finally, what you have read is not a proof. It is a sequence of ideas and diagrams meant to help you understand the ideas behind the proof. I hope this has helped you do that.