From what I know, the hyperbolic trigonometric functions are almost the same as the circular trigonometric functions ($\sin, \cos, \tan$, et cetera without the $h$ suffix), except they output when a line coming from the centre at the given angle hits the surface of a hyperbola, rather than a circle.
From what I've seen, the relevant hyperbola has asymptotes at $45^\circ$, $-135^\circ$, $225^\circ$, and $315^\circ$, so basically the lines $y = 0$ and $x = 0$, but rotated $45^\circ$. This means that a line told to go along any of these asymptotes — defined by the fact that the hyperbola will never meet them — surely will never meet the hyperbola because it basically is the asymptote!?
And anywhere that goes through, judging by the diagrams I've looked at on-line, the empty space above and below the origin (excluding where the hyperbola really is,) should, as well, surely be infinity, since it never gets to the hyperbola?
The summarised question: How come giving a hyperbolic function that treads along the asymptotes does not give infinity or some other undefined answer?