Why can a graph have more than two horizontal asymptotes?
A horizontal asymptote is where the x values go off to infinity in that direction as the y values approach some value. There are only two directions it could go off to infinity – positive and negative. There’s your two. For there to be more would mean the y values would approach two different numbers at the same time. This is not possible. Now, it’s different for a vertical asymptote because they usually represent breaks in the domain, i.e. division by zero. You can have an infinite number of those. But unless we’re talking multivariate/vector calculus, there is a maximum of two horizontal asymptotes. EDIT Jesse is wrong. in order to be a function, the range of y=arctan(x) (or the domain of x=tan(y), as he called it) must be restricted from y = (-pi/2, pi/2) because of the vertical line test, remember? There are still only two asymptotes, sorry.
