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Vertical tangent on the function ƒ(x) at x = c. In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.
The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x) A curve intersecting an asymptote infinitely many times In analytic geometry , an asymptote ( / ˈ æ s ɪ m p t oʊ t / ) of a curve is a line such that the distance between the curve and the line approaches zero as one or ...
The tangent function = / has a simple zero at = and vertical asymptotes at = /, where it has a simple pole of residue . Again, owing to the periodicity, the zeros are all the integer multiples of π {\displaystyle \pi } and the poles are odd multiples of π / 2 {\displaystyle \pi /2} , all having the same residue.
Rational Functions and End Behavior 2 1.8 Rational Functions and Zeros 1 1.9 Rational Functions and Vertical Asymptotes 1 1.10 Rational Functions and Holes 1 1.11 Equivalent Representations of Polynomial and Rational Expressions 2 1.12 Transformations of Functions 2 1.13 Function Model Selection and Assumption Articulation 2 1.14
The asymptotic directions are the same as the asymptotes of the hyperbola of the Dupin indicatrix through a hyperbolic point, or the unique asymptote through a parabolic point. [1] An asymptotic direction is a direction along which the normal curvature is zero: take the plane spanned by the direction and the surface's normal at that point. The ...
In other words, the function has an infinite discontinuity when its graph has a vertical asymptote. An essential singularity is a term borrowed from complex analysis (see below). This is the case when either one or the other limits f ( c − ) {\displaystyle f(c^{-})} or f ( c + ) {\displaystyle f(c^{+})} does not exist, but not because it is ...
Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function x ↦ 1 x {\displaystyle x\mapsto {\frac {1}{x}}} is concave for negative x and convex for positive x , but it has no points of inflection because 0 is not in the domain of the function.
The kappa curve has two vertical asymptotes. In geometry, the kappa curve or Gutschoven's curve is a two-dimensional algebraic curve resembling the Greek letter ϰ (kappa).The kappa curve was first studied by Gérard van Gutschoven around 1662.