The two-sided limit as the denominator approaches zero does not exist in conventional mathematics. I propose a circular number line of infinite radius. As one spins around to the other side, positive infinity is blended into negative infinity.

This is called the real projective line, the special case of projective space over the real numbers in one dimension. It dates back to the 1600s and the work of Desargues.

Oh, here's something I was thinking about that's related to numberlines etc - it sounds like it might be similar to this 'projective line' thing.

Imagine an archer in space - his arrows aren't affected by gravity or airflow, they just go in straight lines - you might consider them as laser arrows if that helps.

He 'stands' on a rotating board.

Next to him, continuing to infinity in each direction is a long wall.

Spin the board and have the archer release arrows at random - now, the randomness of the release of each arrow is limited to 0-360°, but the positions at which the arrows hit the wall (or, for 50% of the time, fly off into unwalled space - but let's ignore those ones) will be 'stretched out' - what is the distribution of the arrows on the wall? I'm thinking something approaching a plot of tan(Ø) (where Ø is the angle at which the release of the arrow occurs) - but - and this is the exciting bit - what you have is a numeric amplifier - your initial value of 0-360 (albeit non ordinal i.e. you can have 15°, 15.1°, 15.0000000001° and all the values in-between) which maps to a position on an infinitely long wall (or number line) OK, so it's not that exciting - I was just trying to make a random number generator at the time, and wanted a way to zoom out from a small array of numbers to a seemingly large one (the cardinality of the 'smaller' set being the limiting factor)