If f(x) is a function and there's some number L with
we draw a dashed horizontal line on the graph at height y = L. This line is called a horizontal asymptote. As x approaches ∞, the function f(x) gets closer to L, in the graph the function gets closer to the dashed horizontal line.
We also draw a horizontal asymptote at y = L if
Then as x approaches -∞, or as we move left on the graph, the function f(x) will approach the dashed horizontal line.
It's fine for a graph to cross over its horizontal asymptote(s). We can have something like this, for example:
The important thing is that as x gets bigger (or more negative), the function is getting closer to the horizontal asymptote.
Finding the horizontal asymptote(s) of a function is the same task as finding the limits of a function f(x) as x approaches ∞ or -∞.
The difference is that horizontal asymptotes are drawn as dashed horizontal lines in a graph, while limits (when they exist) are numbers.