Differential Calculus From Practice to Theory

It should be clear from the sketch below that in addition to the horizontal asymptote \(y=0\text{,}\) the graph of \(y=\frac1x\) also has a vertical asymptote of \(x=0\text{.}\)

We can extend the ideas and the notation we’ve developed for horizontal asymptotes to handle vertical asymptotes as well. First note that since the expression \(\frac1x\) is not defined for \(x=0\) we can’t talk about the behavior of the function at zero. But we can talk about its behavior near zero, and notice the graph is very different depending on whether \(x\lt0\) or \(x\gt0\text{.}\) We will focus first on positive values of \(x\text{.}\)

The limit in equation (12.1) is called a right–hand limit and is read aloud as “the limit as \(x\) approaches zero from the right, of one over \(x\) equals infinity.”

The expression \(x\rightarrow0\) means that \(x\) is approaching the number zero. The qualification \(x\gt0\) specifies that it is approaching from the right. So writing\(\limitX{x}{0}{\frac1x}{x\gt0}=\infty\) means that as \(x\rightarrow0\) from the right, the function \(\frac1x\) increases without bound.

The limit in equation (12.2) is called a left–hand limit and is read aloud as “the limit as \(x\) approaches zero from the left, of one over \(x\) equals negative infinity.” Right- and left-hand limits are generically called one-sided limits. One-sided limits are important theoretical tools but their theoretical value is not important to us right now. We will discuss them in more depth in Section 16.2.

A function \(f(x)\text{,}\) will have a vertical asymptote as \(x\rightarrow a\) if either of the one-sided limits is \(\infty\) or \(-\infty\) (\(f(x)\) is increasing or decreasing without bound). We formalize this statement in the following definition.

In fact, at this point you can probably guess the meanings of the expressions in formula (12.3) without any help from us.