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An asymptote is a straight line that a curve approaches but never meets or crosses. Informally, one may speak of the curve meeting the asymptote "at infinity" although this is not a precise definition. In the equation =, y becomes arbitrarily small in magnitude as x increases.
If x=a is a vertical asymptote of f(x), then x=a+h is a vertical asymptote of f(x-h) If y=c is a horizontal asymptote of f(x), then y=c+k is a horizontal asymptote of f(x)+k; If a known function has an asymptote, then the scaling of the function also have an asymptote. If y=ax+b is an asymptote of f(x), then y=cax+cb is an asymptote of cf(x)
The word horizontal is derived from the Latin horizon, which derives from the Greek ὁρῐ́ζων, meaning 'separating' or 'marking a boundary'. [2] The word vertical is derived from the late Latin verticalis, which is from the same root as vertex, meaning 'highest point' or more literally the 'turning point' such as in a whirlpool.
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 ...
The vertical and horizontal lines are asymptotes. In the same way, it can be shown that the reciprocal of a continuous function r = 1 / f {\displaystyle r=1/f} (defined by r ( x ) = 1 / f ( x ) {\displaystyle r(x)=1/f(x)} for all x ∈ D {\displaystyle x\in D} such that f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} ) is continuous in D ∖ { x : f ...
A sequence of distributions corresponds to a sequence of random variables Z i for i = 1, 2, ..., I . In the simplest case, an asymptotic distribution exists if the probability distribution of Z i converges to a probability distribution (the asymptotic distribution) as i increases: see convergence in distribution.
Secondly, the current lead section, rather than waffling about nonsense like "qualitative" versus "quantitative" properties, should state briefly what a horizontal asymptote, vertical asymptote, and oblique asymptote are. This is the kind of information that would be helpful to readers, not confusing pseudo-pedagogical prose.
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.
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