Asymptote

[1][2] The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means "not falling together", from ἀ priv.

[3] The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.

An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as x tends to +∞ or −∞.

The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem to counter everyday experience.

The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width.

Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.

These ideas are part of the basis of concept of a limit in mathematics, and this connection is explained more fully below.

[6] The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ(x).

These can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on their orientation.

The line x = a is a vertical asymptote of the graph of the function y = ƒ(x) if at least one of the following statements is true: where

The function ƒ(x) may or may not be defined at a, and its precise value at the point x = a does not affect the asymptote.

Moreover, if a function is continuous at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote.

A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.

A function ƒ(x) is asymptotic to the straight line y = mx + n (m ≠ 0) if In the first case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to +∞, and in the second case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to −∞.

The oblique asymptote, for the function f(x), will be given by the equation y = mx + n. The value for m is computed first and is given by where a is either

If this limit doesn't exist then there is no oblique asymptote in that direction.

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 a remainder.

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

[7] From the definition, only open curves that have some infinite branch can have an asymptote.

For example, the upper right branch of the curve y = 1/x can be defined parametrically as x = t, y = 1/t (where t > 0).

A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.

All three types of asymptotes can be present at the same time in specific examples.

An asymptote serves as a guide line to show the behavior of the curve towards infinity.

[13] For example, one may identify the asymptotes to the unit hyperbola in this manner.

A plane algebraic curve is defined by an equation of the form P(x,y) = 0 where P is a polynomial of degree n where Pk is homogeneous of degree k. Vanishing of the linear factors of the highest degree term Pn defines the asymptotes of the curve: setting Q = Pn, if Pn(x, y) = (ax − by) Qn−1(x, y), then the line is an asymptote if

the curve has a singular point at infinity which may have several asymptotes or parabolic branches.

Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches.

, but its highest order term gives the linear factor x with multiplicity 4, leading to the unique asymptote x=0.

The hyperbola has the two asymptotes The equation for the union of these two lines is Similarly, the hyperboloid is said to have the asymptotic cone[16][17] The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.