mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, particularly

calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...

, a vertical tangent is a

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...

that is vertical. Because a vertical line has infinite

slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...

, a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

whose

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

has a vertical tangent is not

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

at the point of tangency.

Limit definition

A function ƒ has a vertical tangent at ''x'' = ''a'' if the

difference quotient In single-variable calculus, the difference quotient is usually the name for the expression : \frac which when taken to the Limit of a function, limit as ''h'' approaches 0 gives the derivative of the Function (mathematics), function ''f''. The ...

used to define the derivative has infinite limit: :

\lim_\frac = \quad\text\quad\lim_\frac = .

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 In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

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

\lim_ f'(x) = \text

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

\lim_ f'(x) = \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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

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_\frac = \quad\text\quad \lim_\frac = \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_ f'(x) =  \quad \text \quad \lim_ f'(x) = \text

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

Example

The function :

f(x) = \sqrt /math>
has a vertical tangent at ''x'' = 0, since it is continuous and
: \lim_ f'(x) \;=\; \lim_ \frac \;=\; \infty. Similarly, the function
: g(x) = \sqrt /math>
has a vertical cusp at ''x'' = 0, since it is continuous,
: \lim_ g'(x) \;=\; \lim_ \frac \;=\; \text and
: \lim_ g'(x) \;=\; \lim_ \frac \;=\; \text{.}

References

Vertical Tangents and Cusps
Retrieved May 12, 2006. Mathematical analysis