Consider, for example, the graph of
As . So the -axis is an asymptote.
But when . So the curve crosses this asymptote at .
As far as your original question is concerned, you would just need to look for solutions to an equation in the usual way. If, for example, the graph of has as an asymptote, then the graph crosses the asymptote at any finite values of that satisfy the equation . Of course, there may not be any such values - but that's another issue!