What is the slope of x=3?

Jun 10, 2015

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 ($m \to \infty$).

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:
$y = m x + q$ 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 $y = 10000 x + 10000$:

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 $A \left(3 , 0\right) \mathmr{and} B \left(3 , 5\right)$ of the line you get this fraction:
${\Delta}_{Y} / {\Delta}_{X} = \frac{5 - 0}{3 - 3} = \frac{5}{0.}$
Obviously this fraction doesn't make sense because it's a particular case.
For this reasons, some people say that $m = \infty$ but it is formally wrong, they should say that $m \to \infty$ because m doesn't exist.