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# Functions, Graphs, and Limits

# Horizontal Asymptoes

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 is 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.