##
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 **
(*x*_{0},*f*(*x*_{0}))** 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
*x*_{0},*f*(*x*_{0})) iff

or

In both cases, *f*'(*x*_{0}) becomes infinite. A graph may also
exhibit a behavior similar to a cusp without having infinite
slopes:

**Example.** Consider the function

*f*(*x*) = |*x*^{3} - 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**.

**Exercise 1.** Does the function

have a vertical tangent or a vertical cusp at *x*=3?
**Answer.**

**Exercise 2.** Does the function

have a vertical tangent or a vertical cusp at *x*=0?
**Answer.**

**
**

**
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