Vertical tangent

In mathematics, particularly calculus, a vertical tangent is a 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 &fnof; has a vertical tangent at x&thinsp;=&thinsp;a if the difference quotient used to define the derivative has infinite limit:


 * $$\lim_{h\to 0}\frac{f(a+h) - f(a)}{h} = {+\infty}\quad\text{or}\quad\lim_{h\to 0}\frac{f(a+h) - f(a)}{h} = {-\infty}.$$

The first case corresponds to an upward-sloping vertical tangent, and the second case to a downward-sloping vertical tangent. The graph of &fnof; has a vertical tangent at x&thinsp;=&thinsp;a if the derivative of &fnof; 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


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

then &fnof; must have an upward-sloping vertical tangent at x&thinsp;=&thinsp;a. Similarly, if


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

then &fnof; must have a downward-sloping vertical tangent at x&thinsp;=&thinsp;a. In these situations, the vertical tangent to &fnof; 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


 * $$\lim_{h \to 0^-}\frac{f(a+h) - f(a)}{h} = {+\infty}\quad\text{and}\quad \lim_{h\to 0^+}\frac{f(a+h) - f(a)}{h} = {-\infty}\text{,}$$

then the graph of &fnof; 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
 * $$\lim_{x \to a^-} f'(x) = {-\infty} \quad \text{and} \quad \lim_{x \to a^+} f'(x) = {+\infty}\text{,}$$

then the graph of &fnof; will have a vertical cusp at x&thinsp;=&thinsp;a that slopes down on the left side and up on the right side.

Example
The function
 * $$f(x) = \sqrt[3]{x}$$

has a vertical tangent at x&thinsp;=&thinsp;0, since it is continuous and
 * $$\lim_{x\to 0} f'(x) \;=\; \lim_{x\to 0} \frac{1}{3\sqrt[3]{x^2}} \;=\; \infty.$$

Similarly, the function
 * $$g(x) = \sqrt[3]{x^2}$$

has a vertical cusp at x&thinsp;=&thinsp;0, since it is continuous,
 * $$\lim_{x\to 0^-} g'(x) \;=\; \lim_{x\to 0^-} \frac{2}{3\sqrt[3]{x}} \;=\; {-\infty}\text{,}$$

and
 * $$\lim_{x\to 0^+} g'(x) \;=\; \lim_{x\to 0^+} \frac{2}{3\sqrt[3]{x}} \;=\; {+\infty}\text{.}$$