Trouble in understanding the result of $\lim\limits_{x \rightarrow 0} \frac{x}{0}$ as shown on Wolfram Alpha

Question

So in wolfram alpha it says that $\lim\limits_{x \to 0} \frac{x}{0} = \infty^\sim$ where $\infty^\sim$ symbol is complex infinity.

I find it hard to understand this symbol and the concept, can someone please explain why this happens and why it is undefined.

From the definition, Complex infinity is an infinite number in the complex plane whose complex argument is unknown or undefined — Andrei, Nov 22 '23 at 15:33
The fact that you can write a math formula (or enter one in Mathematica) does not imply that this formula has any meaning. — Moishe Kohan, Nov 22 '23 at 15:46
In the context of the extended complex plane $\overline{\mathbb{C}}=\mathbb{C}\cup{\infty}$ we have that $\frac{z}{0}=\infty$ for all $z\in\mathbb{C}\setminus{0}$, from which your limit follows. This assumes, however, that we are working in $\overline{\mathbb{C}}$ — Lorago, Nov 22 '23 at 15:53
Slightly related in that it also brings up the "complex infinity" that Wolfram Language team made up: my answer to "Is $\infty+ i\times \infty$: $\tilde{\infty}$?". — Mark S., Nov 23 '23 at 02:45

score 1 · Accepted Answer · answered Nov 22 '23 at 15:40

This happens because we do not know how dividing by 0 looks like and it is simply not defined. You may argue $\lim_{x \rightarrow 0} \frac{x}{x}$ is the same form as you stated. But that is untrue because in your case, we have the fraction a type of $\frac{\rightarrow 0}{exact \;0}$ which is as good as dividing 2 by 0, because in the end $\rightarrow 0$ is still a finite number, just infinitesimally small. And as you do not know what the result of dividing by 0 is, we cannot predict what exactly the outcome may be, imaginary, real or complex

Trouble in understanding the result of $\lim\limits_{x \rightarrow 0} \frac{x}{0}$ as shown on Wolfram Alpha

1 Answers1