As x heads off to positive infinity, f(x) will do the same. This means f(x) grows without bound in the positive x direction.
However, in the other direction, as x goes to negative infinity things are a bit different. Instead of going off to infinity, f(x) will approach y = 0. It will never touch or cross this horizontal line. It will only get closer and closer. Think of it as an electric fence you can't touch.
So there is only one horizontal asymptote and it is the horizontal line y = 0. This is the horizontal line that overlaps the x axis.