We are frequently interested in a function’s “end behavior.” That is, we are interested in how the function behaves 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!
For the function \(f_1\) in the figures from Exercise 9.6.1, we could (correctly) write down that \(\lim\limits_{x \to \infty}\fe{f_1}{x} = -2\) and \(\lim\limits_{x \to -\infty}\fe{f_1}{x} = -2\text{.}\) Go ahead and write (and say aloud) the analogous limits for the functions \(f_2\text{,}\)\(f_3\text{,}\)\(f_4\text{,}\) and \(f_5\text{.}\)
Values of the function \(f\) defined by \(\fe{f}{x} = \frac{3x^2 - 16x + 5}{2x^2 - 13x + 15}\) are shown in the table. Both of the questions below are in reference to this function.
Jorge and Vanessa were in a heated discussion about horizontal asymptotes. Jorge said that functions never cross horizontal asymptotes. Vanessa said Jorge was nuts. Vanessa whipped out her trusty calculator and generated the values in the table to prove her point.
