What is the slope of a vertical line?

The of a vertical line is undefined like ##1/0## is undefined.

If a line passes through distinct points ##(x_1, y_1)## and ##(x_2, y_2)## then the slope of the line is given by the formula:

slope ##m = (Delta y)/(Delta x) = (y_2 - y_1) / (x_2 - x_1)##

If the line is vertical then ##x_2 = x_1## so the denominator is ##0##.

You can mess with the numbers you are using by adding a 'number' called ##oo## which will allow you to express the slope of a vertical line. It can be a useful shorthand, but it does not fix everything and can lead to sloppy reasoning. For example, what is the value of ##0 * oo##?

For a more formal approach to using ##oo## in an advanced setting you might look at the behaviour of

##f(z) = (az+b)/(cz+d)##

on the Riemann sphere ##CC_oo##. Then again, perhaps that's something to look forward to in a few years time.