What is the slope for x = 4?

The definition of slope is for the slope of a line through the points #(x_1, y_1)# and #(x_2, y_2)# with #x_1 != x_2#.

The case #x_1 = x_2#. is not defined.

(You may often hear people say that the slope is infinity. This is the result of a confusion of two or more ideas.)

Remember that the a typical equation of a line can be expressed as

#y=mx+b#

where #m# is the slope of the line. The slope of a line describes ratio of rise (the difference in vertical distance, or #y#-values), divided by the run (the difference in horizontal distance, or #x#-values). In other words, slope can be defined as:

#m=(x_2-x_1)/(y_2-y_1)#

What this means is that as the top part of the fraction gets big (compared to the denominator), the slope gets steeper and steeper, ever creeping closer to a vertical line. Here you have a slope of just 5:

graph{5x+1 [-11.25, 11.26, -5.63, 5.62]}

And here's a slope of 50:

graph{50x+1 [-11.25, 11.26, -5.63, 5.62]}

So the line becomes vertical as #m# gets large. But the equation #x=1# is simply a vertical line at #x=1#. So the slope is #oo#.