We are frequently interested in a function's “end behavior.” That is, what is the behavior of the function as the input variable increases without bound or decreases without bound.

Many times a function will approach a horizontal asymptote as its end behavior. Assuming that the horizontal asymptote \(y=L\) represents the end behavior of the function \(f\) both as \(x\) increases without bound and as \(x\) decreases into the negative without bound, we write \(\lim\limits_{x\to\infty}\fe{f}{x}=L\) and \(\lim\limits_{x\to-\infty}\fe{f}{x}=L\text{.}\)

The formalistic way to read \(\lim\limits_{x\to\infty}\fe{f}{x}=L\) is “the limit of \(\fe{f}{x}\) as \(x\) approaches infinity equals \(L\text{.}\)” When read that way, however, the words need to be taken anything but literally. In the first place, \(x\) isn't approaching anything! The entire point is that \(x\) is increasing without any bound on how large its value becomes. Secondly, there is no place on the real number line called “infinity”; infinity is not a number. Hence \(x\) certainly can't be approaching something that isn't even there!