Question

What is the slope of `x=3`?

Answer 1

It is a degenerated case because`x=3` is not a function. The slope doesn't exist, but we can say that it tends to infinite (`m->oo`).

Explanation:

`x=3` is not a function (there isn't any y, to keep it simpe).
If you take the common line function in space you have:
`y=mx+q` where `m` is the slope.
If you imagine to grow m to infinite you can obtain an almost vertical line. For example see the graph of `y=10000x+10000`:

graph{y=10000x+10000 [-10, 10, -5, 5]}

Anyway `x=k` is a very peculiar case. If you use the common formula to obtain the slope for example for the two points `A(3,0) and B(3,5)` of the line you get this fraction:
`Delta_Y/Delta_X=(5-0)/(3-3)=5/0.`
Obviously this fraction doesn't make sense because it's a particular case.
For this reasons, some people say that `m=oo` but it is formally wrong, they should say that `m->oo` because m doesn't exist.