Vertical tangent

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

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:

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

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

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:

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

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

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

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

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.

References

Vertical Tangents and Cusps. Retrieved May 12, 2006.

Retrieved from "http://en.wikipedia.org/wiki/Vertical_tangent"

Categories: Mathematical analysis

This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.

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