$\lim_{z\to1}\frac{1-z^*}{1-z}$ using Wolfram Alpha

Question

I wanted to verify the answer to many questions on limits of complex numbers.

So, I tried using Wolfram Alpha for the same.

$$\lim_{z\to1}\frac{1-z^*}{1-z}$$ does not exist. [using path $y=m(1-x)$]

But, Wolfram Alpha computes the limit as $1$.

Am I wrong or Wolfram Alpha does not work for complex numbers?

Answer 1

You are right, since$$\lim_{t\to0,\,t\in\Bbb R}\frac{1-\left(\overline{t+1}\right)}{1-(t+1)}=1\quad\text{and}\quad\lim_{t\to0,\,t\in\Bbb R}\frac{1-\left(\overline{ti+1}\right)}{1-(ti+1)}=-1.$$

Answer 2

Let $1-\overline z=re^{i\theta}$, then $1-z=re^{-i\theta}$, so:

$$\lim_{z\to1} \frac{1-\overline z}{1-z}=\lim_{r\to0}\frac{re^{i\theta}}{re^{-i\theta}}=\lim_{r\to0} e^{2i\theta}$$

but $\theta$ is undefined, and so the limit can tend to anything on the unit circle.

Answer 3

Here it is in Maple.

The documentation specifies that if the direction is not specified, then "real bidirectional" is assumed.

Answer 4

Wolfram|Alpha does not always give you access to all the power of Wolfram Language/Mathematica. For this calculation, I can't find an easy way to get it to consider the limit in the complex plane.

But you can use TryItOnline for quick non-graphical calculations like this. For example, the following code outputs Indeterminate, showing that the limit doesn't exist.

Wolfram Language (Mathematica)

Limit[(1 - Conjugate[z])/(1 - z), z -> 1, Direction -> Complexes] // Print

Try it online!