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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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 generalizations ...

calculus
, a vertical tangent is a
tangent line In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

tangent line
that is
vertical Vertical may refer to: * Vertical direction In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects ...
. 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 (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

infinite
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

slope
, a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 ...

graph
has a vertical tangent is not
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ...

differentiable
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 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
, 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 In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέ ...

asymptote
on the graph of the derivative.


Vertical cusps

Closely related to vertical tangents are vertical cusps. This occurs when the
one-sided derivativeIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
s 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