Vertical tangent

In mathematics, a vertical tangent is tangent line with infinite slope, thus being vertical.

Definition

Suppose the function "ƒ"("x") hold the point "P"("c" , "ƒ"("c")). The graph of "ƒ" has a vertical tangent at "P" if one of the following is true:

:$lim\_\{x\; o\; c^+\}\; f\text{'}(x)\; =\; lim\_\{x\; o\; c^-\}\; f\text{'}(x)\; =\; +\; infty$

or

:$lim\_\{x\; o\; c^+\}\; f\text{'}(x)\; =\; lim\_\{x\; o\; c^-\}\; f\text{'}(x)\; =\; -\; infty$

Thus, "ƒ"'("c") = undefined = "m_c", where "m_c" is the slope at "x" = "c".

Vertical asymptotes

A function is able to have a vertical asymptote with no vertical tangent. This occurs when:

:$lim\_\{x\; o\; c^+\}\; f\text{'}(x)\; =\; +\; infty$:$lim\_\{x\; o\; c^-\}\; f\text{'}(x)\; =\; -\; infty$

or

:$lim\_\{x\; o\; c^+\}\; f\text{'}(x)\; =\; -\; infty$:$lim\_\{x\; o\; c^-\}\; f\text{'}(x)\; =\; +\; infty$

As "x" approaches "c", "ƒ"'("x") approaches opposite infinities, resulting in a vertical asymptote; however, because the limits do not approach the same number, a vertical tangent does not exist.

References

[http://www.sosmath.com/calculus/diff/der09/der09.html Vertical Tangents and Cusps] . Retrieved May 12, 2006.