Example4.8.12Undefined Slope

What is the slope of a vertical line? Figure 4.8.13 shows three lines passing through the origin, each steeper than the last. In each graph, you can see a slope triangle that uses a “rise” of \(4\) each time.


If we continued making the line steeper and steeper until it was vertical, the slope triangle would still have a “rise” of \(4\text{,}\) but the “run” would become smaller and smaller, closer to \(0\text{.}\) And then the slope would be \(m=\frac{4}{\text{very small}}=\text{very large}\text{.}\) So the slope of a vertical line can be thought of as “infinitely large.”

If we actually try to compute the slope using the slope triangle when the run is \(0\text{,}\) we would have \(\frac{4}{0}\text{,}\) which is undefined. So we also say that the slope of a vertical line is undefined. Some people say that a vertical line has no slope.