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.

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