Asymptote

An asymptote ( altgr. ἀσύμπτωτος asýmptōtos " non-compliant " AltGr. Πίπτω pípto " I fall " ) is in mathematics a function that approximates another function or a Cartesian axis at infinity.

Asymptote of a curve

The here given representation of asymptotes is more a description than a formal definition of clean.

Curves considered here, the senses are in a sense " one-dimensional " subsets of a Euclidean space, usually the Euclidean plane: Mathematics clean defined examples of such curves are the images of paths, algebraic curves and graphs of continuous functions with countably many definitions gaps ( this applies to most viewed in school functions ). Approaches a straight line on a graph, without the two ever touching the course, then the line an asymptote of the graph.

An asymptote of such a curve is a straight line of the curve " at infinity approximates arbitrarily " itself. More precisely, this means that the distance that has a point of the curve converges to 0 when traveling along the straight line to infinity. Formally, one could write:

Wherein the distance from to is defined as the infimum of the distances of the points by:

For an algebraic curve can be the Asymptotenbegriff from the perspective of projective geometry, as described:

Asymptote of a function

An asymptote is a graph ( for example, a straight line), of the graph of a given function approximates arbitrarily far. Asymptotes of functions are considered in particular in the context of curve sketching.

It has thereby given a function whose domain is a subset of.

One distinguishes between two different types of asymptotes, as a function either - can approach or direction.

Approximately in the y- direction

Has a pole at the point, that is true

Then it is called the line a vertical ( or vertical ) asymptote of or Polgerade of.

Approximately in the x direction

Converges toward to a real number, that is true

Then it is called the line a horizontal ( or horizontal ) asymptote of. The same applies to the limit.

Is a straight line, the applies to the border crossing or approaching any, that is

Then it is called an oblique asymptote of.

These three types of asymptotes together form exactly the asymptotes of the graph, as understood in the sense of the curve of the upper portion " asymptote of the curve ."

The concept of the oblique asymptote is sometimes generalized to take certain lines allow "simple" functions that satisfy the above limit condition ( approximate curves ).

So you can for example allow arbitrary polynomials as an oblique asymptotes. Is a rational function ( polynomial and ) then always has a slant asymptote in this sense. It is the. During polynomial division of resulting polynomial by The vertical distance from on is indicated by the real residual broken rational function which has the same vertical asymptotes as and in addition the horizontal asymptote.

But you can also explain any other classes of functions to oblique asymptotes, if they meet the limit condition. Depending on the application one or the other definition is more appropriate.

Examples

The function ( see hyperbola )

Has the pole, or vertical asymptote at and the horizontal asymptote.

The function

Has in the pole and (if you allow polynomials as an oblique asymptotes ) the approximate parabola.

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