Asymptotic and Parallel Lines Crossing.

[Jim_Peebles]

@markn_1

I think this is getting a bit esoteric for the OP, which was, as I read it, a question about simple asymptotic curves in normal Euclidean space.

Consider the curve y=1/x. As x increases the curve asymptotically approaches the x axis. Do they ever meet? If they did, it would be at a point where y is zero – that is, the point (x,0) would be on the curve for some x. Plugging that into the equation, it says 0=1/x at that point. Clearly there is no such value of x, so the curve never meets the x axis. This is how all asymptotic curves work (that’s the definition of asymptotic). The curve approaches, but never meets, some other curve.

I thought the whole point was to throw out a bunch of esoteric ideas making sense of “parallel lines meet”. As to your “no such value of x”: not if we use hyperreal numbers:

Hyperreal number

Which are employed in “nonstandard analysis” to provide alternate proofs of theorems true in standard analysis (which is restricted to only the usual “real numbers”).