Vertical tangent

In mathematics, 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.

1 Limit definition
2 Vertical cusps
3 Example
4 References

Limit definition

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

$\lim_{h\to 0}\frac{f(a%2Bh) - f(a)}{h} = {%2B\infty}\quad\text{or}\quad\lim_{h\to 0}\frac{f(a%2Bh) - f(a)}{h} = {-\infty}.$

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

$\lim_{x\to a} f'(x) = {%2B\infty}\text{,}$

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

$\lim_{x\to a} f'(x) = {-\infty}\text{,}$

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

$\lim_{h \to 0^-}\frac{f(a%2Bh) - f(a)}{h} = {%2B\infty}\quad\text{and}\quad \lim_{h\to 0^%2B}\frac{f(a%2Bh) - f(a)}{h} = {-\infty}\text{,}$

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

$\lim_{x \to a^-} f'(x) = {-\infty} \quad \text{and} \quad \lim_{x \to a^%2B} f'(x) = {%2B\infty}\text{,}$

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 $\infty$ on the left and $-\infty$ on the right.