I believe the maximum would be two - one when the independent variable tends toward minus infinity, and one when it tends toward plus infinity. Unbounded functions can have lots of asymptotes; for example the periodic tangent function.
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Not sure what non-verticle means, but a rational function can have up to 2 non-vertical asymptotes,
When you graph a tangent function, the asymptotes represent x values 90 and 270.
No, it will always have one.
When you plot a function with asymptotes, you know that the graph cannot cross the asymptotes, because the function cannot be valid at the asymptote. (Since that is the point of having an asymptotes - it is a "disconnect" where the function is not valid - e.g when dividing by zero or something equally strange would occur). So if you graph is crossing an asymptote at any point, something's gone wrong.
Undefined; large