Social Choice and Individual Values

Vertical tangent on the function ƒ(x) at x=c.

In mathematics and Calculus, a vertical tangent is tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.

Limit definition

A function ƒ has a vertical tangent at x = a if the difference quotient used to define the derivative has infinite limit:

$\underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}=+\mathrm{\infty }\text{or}\underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}=-\mathrm{\infty }.$

The first case corresponds to an upward-sloping vertical tangent, and the second case to a downward-sloping vertical tangent. Informally speaking, the graph of ƒ has a vertical tangent at x = a if the derivative of ƒ at a is either positive or negative infinity.

For a continuous function, it is often possible to detect a vertical tangent by taking the limit of the derivative. If

$\underset{x\to a}{lim}{f}^{\prime }(x)=+\mathrm{\infty }\text{,}$

then ƒ must have an upward-sloping vertical tangent at x = a. Similarly, if

$\underset{x\to a}{lim}{f}^{\prime }(x)=-\mathrm{\infty }\text{,}$

then ƒ must have a downward-sloping vertical tangent at x = a. In these situations, the vertical tangent to ƒ appears as a vertical asymptote on the graph of the derivative.

Vertical cusps

Closely related to vertical tangents are vertical cusps. This occurs when the one-sided derivatives are both infinite, but one is positive and the other is negative. For example, if

$\underset{h\to 0^{-}}{lim}\frac{f(a+h)-f(a)}{h}=+\mathrm{\infty }\text{and}\underset{h\to 0^{+}}{lim}\frac{f(a+h)-f(a)}{h}=-\mathrm{\infty }\text{,}$

then the graph of ƒ will have a vertical cusp that slopes up on the left side and down on the right side.

As with vertical tangents, vertical cusps can sometimes be detected for a continuous function by examining the limit of the derivative. For example, if

$\underset{x\to a^{-}}{lim}{f}^{\prime }(x)=-\mathrm{\infty }\text{and}\underset{x\to a^{+}}{lim}{f}^{\prime }(x)=+\mathrm{\infty }\text{,}$

then the graph of ƒ will have a vertical cusp that slopes down on the left side and up on the right side. This corresponds to a vertical asymptote on the graph of the derivative that goes to $\mathrm{\infty }$ on the left and $-\mathrm{\infty }$ on the right.

Example

The function

$f(x)=\sqrt[3]{x}$

has a vertical tangent at x = 0, since it is continuous and

$\underset{x\to 0}{lim}{f}^{\prime }(x)=\underset{x\to 0}{lim}\frac{1}{\sqrt[3]{x^2}}=\mathrm{\infty }.$

Similarly, the function

$g(x)=\sqrt[3]{x^2}$

has a vertical cusp at x = 0, since it is continuous,

$\underset{x\to 0^{-}}{lim}{g}^{\prime }(x)=\underset{x\to 0^{-}}{lim}\frac{1}{\sqrt[3]{x}}=-\mathrm{\infty }\text{,}$

and

$\underset{x\to 0^{+}}{lim}{g}^{\prime }(x)=\underset{x\to 0^{+}}{lim}\frac{1}{\sqrt[3]{x}}=+\mathrm{\infty }\text{.}$

References

Vertical Tangents and Cusps. Retrieved May 12, 2006.