Vertical tangent - Citizendia

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

In mathematics, a vertical tangent is tangent line with infinite slope, thus being vertical. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Slope is used to describe the steepness incline gradient or grade of a straight line.

Definition

Suppose the function ƒ(x) hold the point P(c , ƒ(c)). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function 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. In Mathematics, defined and undefined are used to explain whether or not expressions have meaningful sensible and unambiguous values

Vertical asymptotes

A function is able to have a vertical asymptote with no vertical tangent. An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity 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.

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