Vertical Tangents and Cusps

In the definition of the slope, vertical lines were excluded. It is customary not to assign a slope to these lines. This is true as long as we assume that a slope is a number. But from a purely geometric point of view, a curve may have a vertical tangent. Think of a circle (with two vertical tangent lines). We still have an equation, namely x=c, but it is not of the form y = ax+b. In fact, such tangent lines have an infinite slope. To be precise we will say:
The graph of a function f(x) has a vertical tangent at the point (x₀,f(x₀)) if and only if 

Example. Consider the function

We have

Clearly, f'(2) does not exist. In fact we have

So the graph of f(x) has a vertical tangent at (2,0). The equation of this line is x=2.


In this example, the limit of f'(x) when  is the same whether we get closer to 2 from the left or from the right. In many examples, that is not the case.
Example. Consider the function

We have

So we have

It is clear that the graph of this function becomes vertical and then virtually doubles back on itself. Such pattern signals the presence of what is known as a vertical cusp. In general we say that the graph of f(x) has a vertical cusp at x₀,f(x₀)) iff

or

In both cases, f'(x₀) becomes infinite. A graph may also exhibit a behavior similar to a cusp without having infinite slopes:
Example. Consider the function

f(x) = |x³ - 8|.

Clearly we have

Hence

Direct calculations show that f'(2) does not exist. In fact, we have left and right derivatives with

So there is no vertical tangent and no vertical cusp at x=2. In fact, the phenomenon this function shows at x=2 is usually called a corner.