Validity and precedent of considering the real number line as a circle where positive infinity and negative infinity are the same number

Question

It occurred to me that there are some situations (e.g. division by 0 and the tan function) where the two limits of a value, approached from below and above, are negative infinity and positive infinity. And that perhaps instead of viewing negative infinity and positive infinity as two different values/numbers, with the limit not existing in these situations, we could see them as the same number at one end of a circle with zero on one side and positive/negative infinity on the opposite side. I'm curious as to the utility or validity of this approach, and if it has any precedent.

Answer 1

Yes, this has been considered; the most common name for it which captures its properties and will help you look up more information is the "projective real line" (as opposed to the extended real line with both $\infty$ and $-\infty$).

But if you don't care about geometry or algebra and just the vague effect this has on shapes, then topologically, this would be the Alexandroff/one-point compactification of the real line (as opposed to the Freudenthal/end compactification).