Yes, I've read a number of definitions
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.
And for most examples, such as $y=1/x$, it seems clear what's going on, and how to discuss the asymptote. Yesterday, a student asked me about this function -
Only after I told her it was a discontinuous function, and when drawing it, be certain to show that at x=0, both lines should be an open circle indicating "not a point", did she tell me the teacher said that $y=1$ and $y=-1$ were asymptotes. I think there's a mistake here as there's nothing approaching the lines, the lines are the equation. Just as a first order binomial, $y=x+2$, is just a line, no asymptotes there.
This prompted the follow on -
And the question of whether today's usage allows crossing to occur. In this example, the distance itself, from first definition, will cross zero, infinitely many times. Looking at the quoted definition, what does "modern authors" mean? Do I need to be concerned that an older book will call the X axis of second graph "Not Asymptote", but a newer one is fine?