Why doesn't x/0 = ±∞

Question

I was watching a video on numberphile about dividing by 0 and It said that x/0=Undefined or Error since it could be + or - ∞. Question is why did mathematicians call it Undefined instead of ±∞? This way if we did 20/0 on a calculator it would give us ±∞ instead of Error.

Also if you said to someone that x/0=±∞ would you be right or wrong?

Answer 1

According to that video, the method utilized holds that if you approach for the right you get $+\infty$ but if you approch for the left you get $-\infty$. If we define this two terms (because infinity is not a number, is a concept) as different (infinity to the left is different to infinity to the right) you get two different answers. And you can't have two different answers for division, because division is an operator defined to have only one answer.

Answer 2

$\pm \infty$ is:
i) not a real number
ii) two concepts.
If $\frac{20}{0}$ were to yield two "answers", neither of which were real numbers, would that match your intuition as to what it means to have a defined value?