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A locally connected curve A is said to be an asymptote of the locally connected curve B when the following is true: Image File history File links Question_mark. ... Asymptote can refer to: Asymptote: A mathematical concept Asymptote (vector graphics language) Asymptote (architects), an architectural design firm This is a disambiguation page: a list of articles associated with the same title. ...

For any positive ε, there exist unbounded connected subsets (pieces of the respective curves) and , such that for every point in its distance to the nearest point in is lower than ε.

In other words, as one moves along B in some direction, the distance between it and the asymptote A eventually becomes smaller than any distance that one may specify. In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ...

If a curve A has the curve B as an asymptote, one says that A is asymptotic to B. Similarly B is asymptotic to A, so A and B are called asymptotic.

Basically, a linear asymptote is straight line which a curve approaches infinitesimally closely, but never exactly meets the curve - hence both the curve and the straight line both extend infinitely. Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... For other uses, see Infinity (disambiguation). ...

Asymptotes, graphs and definitions

Meaning

graphed on Cartesian coordinates. The x and y axes are the asymptotes.

Asymptotes are formally defined using limits. There are many different cases that can be treated separately, such as linear asymptotes (below), although intuitively the two functions become arbitrarily close. Image File history File links Hyperbola_one_over_x. ... Image File history File links Hyperbola_one_over_x. ... In mathematics, the limit of a function is a fundamental concept in analysis. ...

A specific example of linear asymptotes can be found in the graph of the function f(x) = 1/x, in which two asymptotes are seen: the horizontal line y = 0 and the vertical line x = 0. Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

There are multiple ways of interpreting asymptotic behavior. In particular the statement "A function f(x) is said to be asymptotic to a function g(x) as x → ∞" has any of at least three distinct meanings:

1. f(x) − g(x) → 0.
2. f(x) / g(x) → 1.
3. f(x) / g(x) has a nonzero limit.

More formally, curves A and B are asymptotic if and only if there exist continuous functions , such that all of the following conditions are all true:

Multiple asymptotes, intersection

The graph of a function can have vertical, horizontal and slant asymptotes, e.g. y = x ^{| x | / x} + 1 / x.

A curve can intersect its asymptote, even infinitely many times.

A function may have multiple asymptotes, of different or the same kind. One such function with a horizontal, vertical, and oblique asymptote is graphed to the right above. Image File history File links Asymptote02. ... Image File history File links Asymptote02. ...

In particular a function y = ƒ(x) can have at most 2 horizontal or 2 oblique asymptotes (or one of each). There may be any number of vertical asymptotes, such as y=tan(x)

A curve may cross its asymptote repeatedly or may never actually coincide with it. A curve may have multiple asymptotes. Further, it may even intersect an asymptote infinitely many times, as graphed to the left.

Linear asymptotes

Horizontal asymptotes

The graph of a function can have two horizontal asymptotes. An example of such a function would be y = arctan(x).

Suppose f is a function. Then the line y = a is a horizontal asymptote for f if Image File history File links Asymptote03. ...

Intuitively, this means that f(x) can be made as close as desired to a by making x big enough. How big is big enough depends on how close one wishes to make f(x) to a. This means that far out on the curve, the curve will be close to the line.

Note that if

then the graph of f has two horizontal asymptotes: y = a and y = b. An example of such a function is the arctangent function.

Another example would be ƒ(x)=1/(x²+1), which has a horizontal asymptote at y=0, as can be seen by the limit

Vertical asymptotes

The line x = a is a vertical asymptote of a function f if either of the following conditions is true:

Intuitively, if x = a is an asymptote of f, then, if we imagine x approaching a from one side, the value of f(x) grows without bound; i.e., f(x) becomes large (positively or negatively), and, in fact, becomes larger than any finite value.

Note that f(x) may or may not be defined at a: what the function is doing precisely at x = a does not affect the asymptote. For example, consider the function

As , f(x) has a vertical asymptote at 0, even though f(0) = 5.

Another example is ƒ(x) = 1/(x-1) which has a vertical asymptote of x=1 as shown by the limit

Oblique asymptotes

In the graph of , the y-axis (x = 0) and the line y = x are both asymptotes.

When a linear asymptote is not parallel to the x- or y-axis, it is called either an oblique asymptote or slant asymptote. the function f(x) is asymptotic to y = mx + b if

Note that y = mx + b is never a vertical asymptote, but can be a horizontal asymptote if m=0.

An example is ƒ(x)=(x²-1)/x which has an oblique asymptote of y=x (m=1, b=0) as seen in the limit

Computationally identifying an oblique asymptote can be more difficult than a horizontal or vertical asymptote, in particular because the m and b might not be known. It is typical to evaluate the appropriate limit and choose m, b so that it exists. For example, to find the oblique asymptote of y=25(x³+2x²+3x+4)/(5x²+6x+7), one can evaluate the limit

So the oblique asymptote is y=5x+4.

Nonlinear asymptotes

Curves may be asymptotic to each other without either being linear. In this case the general characterizations are typically necessary. For example, (x³+2x²+3x+4)/(x) is asymptotic to x²+2x+3 because of the limit

Also, (e^x)/(2x+1) is asymptotic to (e^x)/x because of the limit

However, e^x is not asymptotic to (e^x)/x because of the limit

Elementary methods for identifying linear asymptotes

Asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits).

Rational functions

A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes. In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

The degree of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator. The degree of a polynomial is the maximum of the degrees of all terms in the polynomial. ...

Table listing the cases of horizontal and oblique asymptotes for rational functions

deg(numerator) - deg(denominator)	Horizontal/oblique asymptotes	Example, asymptote
<0	y=0
0	y="ratio of leading coefficients"
1	1 oblique
>1	None

The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at x=0, and x=1, but not at x=2

Oblique asymptotes

Black: the graph of . Red: the asymptote y = x. Green: difference between the graph and it's asymptote for x = 1,2,3,4,5,6

When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after dividing the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and an error term. For example, consider the function In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of lower degree, a generalized version of the familiar arithmetic technique called long division. ...

shown to the right. As the value of x increases, f approaches the asymptote y=x. This is because the other term, y=1/(x+1) becomes smaller.

If the degree of the numerator is more than 1 larger than the degree of the denominator, there will generally still be an error term that goes to zero as x increases, but the quotient will not be linear, and the function does not have an oblique asymptote.

The error term need not be so simple, however, as in this example.

Translations of known functions

If a known function has an asymptote (such as y=0 for f(x)=e^x), then the translations of it also have an asymptote.

• If x=a is a vertical asymptote of f(x), then x=a+k is a vertical asymptote of f(x-h)+k
• If y=b is a horizontal asymptote of f(x), then y=b+h is a horizontal asymptote of f(x-h)+k

For example, f(x)=e^x-1+2 has horizontal asymptote y=0+2=2, and no vertical or oblique asymptotes.

See also

• Asymptotic analysis
• Asymptotic curve
• Big O notation