A horizontal asymptote is a fixed value that a function approaches as #x# becomes very large in either the positive or negative direction. That is, for a function #f(x)#, the horizontal asymptote will be equal to #lim_(x->+-infty)f(x)#.
As the size of #x# increases to very large values (i.e. approaches #infty#), functions behave in different ways. Some simply get bigger and bigger forever (e.g. #x^2#). Others oscillate up and down (e.g. #sinx#). But others get closer and closer to a particular constant value. For instance, consider the hyperbola #y=1/x#.
When we input very large values of #x# (think of #x=1000#, #x=1000000#, #x=1000000000# etc.), the value of #1/x# becomes very small. It gets very close to #0#. And if we plot points to see this visually, we find that the graph of #y=1/x# approaches the line #y=0#. We can observe the same effect when putting in very large negative values of #x#, too. So we say that #y=0# is the horizontal asymptote of #y=1/x#.
If we shifted the entire graph upward, say #y=1/x+5#, then we would find that the graph approaches the line #y=5# instead. That's because #lim_(x->+-infty)1/x=0# and so all that's "left over" when considering the horizontal asymptote is the #5#.
A couple of important notes. Firstly, even though the graph approaches the asymptote, it will never get there. There is no value of #x# we can input into #y=1/x# that will make #y# actually equal to #0#! So even though the graph approaches a certain value (like #x=0#), it's critical to understand that "asymptotic behaviour" means you never actually get there.
Secondly, horizontal asymptotes tell you how a graph behaves at its extremities (i.e. as #x->+-infty#). It doesn't tell you anything about how the graph behaves "in the middle", near the #y#-axis. For instance, going back to our example of #y=1/x#, the graph has nothing to do with #y=0# for small values of #x#. In some instances, such as #y=x/(x^2+1)#, the graph can even cross the horizontal asymptote. This is a critical difference between horizontal and vertical asymptotes! (Vertical asymptotes are a different thing entirely, even though they look very similar.)
