Vertical tangent

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Vertical tangent on the function ƒ(x) at x=c.
Vertical tangent on the function ƒ(x) at x=c.

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

[edit] 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 \to c^+} f'(x) = \lim_{x \to c^-} f'(x) = + \infty

or

\lim_{x \to c^+} f'(x) = \lim_{x \to c^-} f'(x) = - \infty


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

[edit] Vertical asymptotes

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

\lim_{x \to c^+} f'(x) = + \infty
\lim_{x \to c^-} f'(x) = - \infty

or

\lim_{x \to c^+} f'(x) = - \infty
\lim_{x \to c^-} f'(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.

[edit] References

Vertical Tangents and Cusps. Retrieved May 12, 2006.