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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 generalizati ...
, a vertical tangent is a
tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
that is
vertical Vertical is a geometric term of location which may refer to: * Vertical direction, the direction aligned with the direction of the force of gravity, up or down * Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
. Because a vertical line has
infinite Infinite may refer to: Mathematics * Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
, 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-oriente ...
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 discre ...
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 its ...
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 as ''h'' approaches 0 gives the derivative of the function ''f''. The name of the expression stems from the fact ...
used to define the derivative has infinite limit: :\lim_\frac = \quad\text\quad\lim_\frac = . The first case corresponds to an upward-sloping vertical tangent, and the second case to a downward-sloping vertical tangent. 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 continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
, 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 context ...
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. This corresponds to a vertical asymptote on the graph of the derivative that goes to \infty on the left and -\infty on the right.


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