A horizontal asymptote is equivalent to the long term behavior of x as x approaches positive and negative #oo#. As #x->oo, or x->-oo #, the function #x/(x^2-1)# approaches 0.
We can tell this is the case by a simple graph or by reasoning that in the grand scheme of things #x^2-1~~x^2#. Imagine you wanted #1000# groups of #1000# cookies. #1000^2# would be the amount of cookies needed, but let's say you only have #1000^2-1#.
The results of the two equations are #1000000# and #999999#. If we had #1000000# people each eat one cookie, only one of them would be unhappy, because they don't have a cookie. Meaning #999999# of them would be happy. That's a 99.9999% success rate. As the number of groups and cookies grow, the success rate grows closer and closer to #100%#, so we can essentially assume that it equivalent in the case of an asymptote.
This allows us to simplify the equation #x/(x^2+1)# to #x/x^2=1/x#
As x grows infinitely large or small, 1/x will get closer and closer to 0, making the asymptote on each side #x=0#.
