The thing is that there is no asymptote in the naming issue. There is no "limit as x goes to infinity" here. There's just a dual name.

Anyone who can do "long" division can show that 1/3 = 0.333 (repeating 3).
My examples above don't involve any functions, just simple grade-school math.

It is funny and counter-intuitive that rational numbers should have 2 names, but it doesn't really change anything. It's just unexpected, and cool.