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Asymptote

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From Wikipedia, the free encyclopedia

Limit of the tangent line at a point that tends to infinity

For other uses, seeAsymptote (disambiguation).

"Asymptotic" redirects here; not to be confused withAsymptomatic.

The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given byy = 2x)

A curve intersecting an asymptote infinitely many times

Inanalytic geometry, anasymptote (/ˈæsɪmptoʊt/) of acurve is a line such that the distance between the curve and the line approaches zero as one or both of thex ory coordinatestends to infinity. Inprojective geometry and related contexts, an asymptote of a curve is a line which istangent to the curve at apoint at infinity.^[1]^[2]

The word asymptote is derived from theGreek ἀσύμπτωτος (asumptōtos) which means "not falling together", from ἀpriv. + σύν "together" + πτωτ-ός "fallen".^[3] The term was introduced byApollonius of Perga in his work onconic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.^[4]

There are three kinds of asymptotes:horizontal,vertical andoblique. For curves given by thegraph of afunctiony =ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches asx tends to+∞ or −∞. Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it asx tends to+∞ or −∞.

More generally, one curve is acurvilinear asymptote of another (as opposed to alinear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although the termasymptote by itself is usually reserved for linear asymptotes.

Asymptotes convey information about the behavior of curvesin the large, and determining the asymptotes of a function is an important step in sketching its graph.^[5] The study of asymptotes of functions, construed in a broad sense, forms a part of the subject ofasymptotic analysis.

Introduction

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$f(x)={\tfrac {1}{x}}$ graphed onCartesian coordinates. Thex andy-axis are the asymptotes.

{\displaystyle f(x)={\tfrac {1}{x}}} — $f(x)={\tfrac {1}{x}}$ graphed onCartesian coordinates. Thex andy-axis are the asymptotes.

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. So if they were to be extended far enough they would seem to merge, at least as far as the eye could discern. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (seeLine). Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.

Consider the graph of the function $f(x)={\frac {1}{x}}$ shown in this section. The coordinates of the points on the curve are of the form $\left(x,{\frac {1}{x}}\right)$ where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5), (5, 0.2), (10, 0.1), ... As the values of $x {\displaystyle x}$ become larger and larger, say 100, 1,000, 10,000 ..., putting them far to the right of the illustration, the corresponding values of $y {\displaystyle y}$ , .01, .001, .0001, ..., become infinitesimal relative to the scale shown. But no matter how large $x {\displaystyle x}$ becomes, its reciprocal ${\frac {1}{x}}$ is never 0, so the curve never actually touches thex-axis. Similarly, as the values of $x {\displaystyle x}$ become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of $y {\displaystyle y}$ , 100, 1,000, 10,000 ..., become larger and larger. So the curve extends further and further upward as it comes closer and closer to they-axis. Thus, both thex andy-axis are asymptotes of the curve. These ideas are part of the basis of concept of alimit in mathematics, and this connection is explained more fully below.^[6]

Asymptotes of functions

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The asymptotes most commonly encountered in the study ofcalculus are of curves of the formy =ƒ(x). These can be computed usinglimits and classified intohorizontal,vertical andoblique asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches asx tends to +∞ or −∞. As the name indicates they are parallel to thex-axis. Vertical asymptotes are vertical lines (perpendicular to thex-axis) near which the function grows without bound. Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 asx tends to +∞ or −∞.

Vertical asymptotes

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The linex =a is avertical asymptote of the graph of the functiony =ƒ(x) if at least one of the following statements is true:

$\lim _{x\to a^{-}}f(x)=\pm \infty ,$
$\lim _{x\to a^{+}}f(x)=\pm \infty ,$

where $\lim _{x\to a^{-}}$ is the limit asx approaches the valuea from the left (from lesser values), and $\lim _{x\to a^{+}}$ is the limit asx approachesa from the right.

For example, if ƒ(x) =x/(x–1), the numerator approaches 1 and the denominator approaches 0 asx approaches 1. So

\lim _{x\to 1^{+}}{\frac {x}{x-1}}=+\infty

\lim _{x\to 1^{-}}{\frac {x}{x-1}}=-\infty

and the curve has a vertical asymptotex = 1.

The functionƒ(x) may or may not be defined ata, and its precise value at the pointx =a does not affect the asymptote. For example, for the function

f(x)={\begin{cases}{\frac {1}{x}}&{\text{if }}x>0,\\5&{\text{if  }}x\leq 0.\end{cases}}

has a limit of +∞ asx → 0⁺,ƒ(x) has the vertical asymptotex = 0, even thoughƒ(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (ora vertical line in general) in more than one point. Moreover, if a function iscontinuous 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.

If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is

f(x)={\tfrac {1}{x}}+\sin({\tfrac {1}{x}})\quad

\quad x=0

This function has a vertical asymptote at $x=0,$ because

\lim _{x\to 0^{+}}f(x)=\lim _{x\to 0^{+}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=+\infty ,

and

\lim _{x\to 0^{-}}f(x)=\lim _{x\to 0^{-}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=-\infty

The derivative of $f {\displaystyle f}$ is the function

f'(x)={\frac {-(\cos({\tfrac {1}{x}})+1)}{x^{2}}}

For the sequence of points

x_{n}={\frac {(-1)^{n}}{(2n+1)\pi }},\quad

for

\quad n=0,1,2,\ldots

that approaches $x=0$ both from the left and from the right, the values $f'(x_{n})$ are constantly $0 {\displaystyle 0}$ . Therefore, bothone-sided limits of $f^{'} {\displaystyle f'}$ at $0 {\displaystyle 0}$ can be neither $+\infty$ nor $-\infty$ . Hence $f'(x)$ doesn't have a vertical asymptote at $x=0$ .

Horizontal asymptotes

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Thearctangent function has two different asymptotes.

Horizontal asymptotes are horizontal lines that the graph of the function approaches asx → ±∞. The horizontal liney = c is a horizontal asymptote of the functiony = ƒ(x) if

\lim _{x\rightarrow -\infty }f(x)=c

\lim _{x\rightarrow +\infty }f(x)=c

In the first case,ƒ(x) hasy = c as asymptote whenx tends to−∞, and in the secondƒ(x) hasy = c as an asymptote asx tends to+∞.

For example, thearctangent function satisfies

\lim _{x\rightarrow -\infty }\arctan(x)=-{\frac {\pi }{2}}

and

\lim _{x\rightarrow +\infty }\arctan(x)={\frac {\pi }{2}}.

So the liney = –π/2 is a horizontal asymptote for the arctangent whenx tends to–∞, andy =π/2 is a horizontal asymptote for the arctangent whenx tends to+∞.

Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the functionƒ(x) = 1/(x²+1) has a horizontal asymptote aty = 0 whenx tends both to−∞ and+∞ because, respectively,

\lim _{x\to -\infty }{\frac {1}{x^{2}+1}}=\lim _{x\to +\infty }{\frac {1}{x^{2}+1}}=0.

Other common functions that have one or two horizontal asymptotes includex ↦ 1/x (that has anhyperbola as it graph), theGaussian function $x\mapsto \exp(-x^{2}),$ theerror function, and thelogistic function.

Oblique asymptotes

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In the graph of $f(x)=x+{\tfrac {1}{x}}$ , they-axis (x = 0) and the liney =x are both asymptotes.

{\displaystyle f(x)=x+{\tfrac {1}{x}}} — In the graph of $f(x)=x+{\tfrac {1}{x}}$ , they-axis (x = 0) and the liney =x are both asymptotes.

When a linear asymptote is not parallel to thex- ory-axis, it is called anoblique asymptote orslant asymptote. A functionƒ(x) is asymptotic to the straight liney =mx +n (m ≠ 0) if

\lim _{x\to +\infty }\left[f(x)-(mx+n)\right]=0\,{\mbox{ or }}\lim _{x\to -\infty }\left[f(x)-(mx+n)\right]=0.

In the first case the liney =mx +n is an oblique asymptote ofƒ(x) whenx tends to +∞, and in the second case the liney =mx +n is an oblique asymptote ofƒ(x) whenx tends to −∞.

An example isƒ(x) = x + 1/x, which has the oblique asymptotey = x (that ism = 1,n = 0) as seen in the limits

\lim _{x\to \pm \infty }\left[f(x)-x\right]

=\lim _{x\to \pm \infty }\left[\left(x+{\frac {1}{x}}\right)-x\right]

=\lim _{x\to \pm \infty }{\frac {1}{x}}=0.

Elementary methods for identifying asymptotes

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

General computation of oblique asymptotes for functions

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The oblique asymptote, for the functionf(x), will be given by the equationy =mx +n. The value form is computed first and is given by

m\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}f(x)/x

wherea is either $-\infty$ or $+\infty$ depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction.

Havingm then the value forn can be computed by

n\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}(f(x)-mx)

wherea should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit definingm exist. Otherwisey =mx +n is the oblique asymptote ofƒ(x) asx tends toa.

For example, the functionƒ(x) = (2x² + 3x + 1)/x has

m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {2x^{2}+3x+1}{x^{2}}}=2

and then

n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\left({\frac {2x^{2}+3x+1}{x}}-2x\right)=3

so thaty = 2x + 3 is the asymptote ofƒ(x) whenx tends to +∞.

The functionƒ(x) = ln x has

m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {\ln x}{x}}=0

and then

n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\ln x

, which does not exist.

Soy = ln x does not have an asymptote whenx tends to +∞.

Asymptotes for rational functions

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Arational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.

Thedegree 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 cases of horizontal and oblique asymptotes for rational functions
deg(numerator)−deg(denominator)	Asymptotes in general	Example	Asymptote for example
< 0	$y=0$	$f(x)={\frac {1}{x^{2}+1}}$	$y=0$
= 0	y = the ratio of leading coefficients	$f(x)={\frac {2x^{2}+7}{3x^{2}+x+12}}$	$y={\frac {2}{3}}$
= 1	y = the quotient of theEuclidean division of the numerator by the denominator	$f(x)={\frac {2x^{2}+3x+5}{x}}=2x+3+{\frac {5}{x}}$	$y=2x+3$
> 1	none	$f(x)={\frac {2x^{4}}{3x^{2}+1}}$	no linear asymptote, but acurvilinear asymptote exists

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 atx = 0, andx = 1, but not atx = 2.

f(x)={\frac {x^{2}-5x+6}{x^{3}-3x^{2}+2x}}={\frac {(x-2)(x-3)}{x(x-1)(x-2)}}

Oblique asymptotes of rational functions

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Black: the graph of $f(x)=(x^{2}+x+1)/(x+1)$ . Red: the asymptote $y=x$ . Green: difference between the graph and its asymptote for $x=1,2,3,4,5,6$ .

{\displaystyle f(x)=(x^{2}+x+1)/(x+1)} — Black: the graph of $f(x)=(x^{2}+x+1)/(x+1)$ . Red: the asymptote $y=x$ . Green: difference between the graph and its 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 afterdividing the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example, consider the function

f(x)={\frac {x^{2}+x+1}{x+1}}=x+{\frac {1}{x+1}}

shown to the right. As the value ofx increases,f approaches the asymptotey =x. This is because the other term, 1/(x+1), approaches 0.

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 asx increases, but the quotient will not be linear, and the function does not have an oblique asymptote.

Transformations of known functions

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

Ifx=a is a vertical asymptote off(x), thenx=a+h is a vertical asymptote off(x-h)
Ify=c is a horizontal asymptote off(x), theny=c+k is a horizontal asymptote off(x)+k

If a known function has an asymptote, then thescaling of the function also have an asymptote.

Ify=ax+b is an asymptote off(x), theny=cax+cb is an asymptote ofcf(x)

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

General definition

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(sec(t), cosec(t)), or x² + y² = (xy)², with 2 horizontal and 2 vertical asymptotes

LetA : (a,b) →R² be aparametric plane curve, in coordinatesA(t) = (x(t),y(t)). Suppose that the curve tends to infinity, that is:

\lim _{t\rightarrow b}(x^{2}(t)+y^{2}(t))=\infty .

A line ℓ is an asymptote ofA if the distance from the pointA(t) to ℓ tends to zero ast → b.^[7] From the definition, only open curves that have some infinite branch can have an asymptote. No closed curve can have an asymptote.

For example, the upper right branch of the curvey = 1/x can be defined parametrically asx = t,y = 1/t (wheret > 0). First,x → ∞ ast → ∞ and the distance from the curve to thex-axis is 1/t which approaches 0 ast → ∞. Therefore, thex-axis is an asymptote of the curve. Also,y → ∞ ast → 0 from the right, and the distance between the curve and they-axis ist which approaches 0 ast → 0. So they-axis is also an asymptote. A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.

Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is $ax+by+c=0$ then the distance from the pointA(t) = (x(t),y(t)) to the line is given by

{\frac {|ax(t)+by(t)+c|}{\sqrt {a^{2}+b^{2}}}}

if γ(t) is a change of parameterization then the distance becomes

{\frac {|ax(\gamma (t))+by(\gamma (t))+c|}{\sqrt {a^{2}+b^{2}}}}

which tends to zero simultaneously as the previous expression.

An important case is when the curve is thegraph of areal function (a function of one real variable and returning real values). The graph of the functiony = ƒ(x) is the set of points of the plane with coordinates (x,ƒ(x)). For this, a parameterization is

t\mapsto (t,f(t)).

This parameterization is to be considered over the open intervals (a,b), wherea can be −∞ andb can be +∞.

An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation isx = c, for some real numberc. The non-vertical case has equationy =mx +n, wherem and $n {\displaystyle n}$ are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.

Curvilinear asymptotes

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x²+2x+3 is a parabolic asymptote to (x³+2x²+3x+4)/x.

LetA : (a,b) →R² be a parametric plane curve, in coordinatesA(t) = (x(t),y(t)), andB be another (unparameterized) curve. Suppose, as before, that the curveA tends to infinity. The curveB is a curvilinear asymptote ofA if the shortest distance from the pointA(t) to a point onB tends to zero ast → b. SometimesB is simply referred to as an asymptote ofA, when there is no risk of confusion with linear asymptotes.^[8]

For example, the function

y={\frac {x^{3}+2x^{2}+3x+4}{x}}

has a curvilinear asymptotey =x² + 2x + 3, which is known as aparabolic asymptote because it is aparabola rather than a straight line.^[9]

Asymptotes and curve sketching

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Asymptotes are used in procedures ofcurve sketching. An asymptote serves as a guide line to show the behavior of the curve towards infinity.^[10] In order to get better approximations of the curve, curvilinear asymptotes have also been used^[11] although the termasymptotic curve seems to be preferred.^[12]

Algebraic curves

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Acubic curve,the folium of Descartes (solid) with a single real asymptote (dashed)

The asymptotes of analgebraic curve in theaffine plane are the lines that are tangent to theprojectivized curve through apoint at infinity.^[13] For example, one may identify theasymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves,^[14] although they also make sense when defined in this way for curves over an arbitraryfield.^[15]

A plane curve of degreen intersects its asymptote at most atn−2 other points, byBézout's theorem, as the intersection at infinity is of multiplicity at least two. For aconic, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.

A plane algebraic curve is defined by an equation of the formP(x,y) = 0 whereP is a polynomial of degreen

P(x,y)=P_{n}(x,y)+P_{n-1}(x,y)+\cdots +P_{1}(x,y)+P_{0}

whereP_k ishomogeneous of degreek. Vanishing of the linear factors of the highest degree termP_n defines the asymptotes of the curve: settingQ =P_n, ifP_n(x,y) = (ax −by)Q_n−1(x,y), then the line

Q'_{x}(b,a)x+Q'_{y}(b,a)y+P_{n-1}(b,a)=0

is an asymptote if $Q'_{x}(b,a)$ and $Q'_{y}(b,a)$ are not both zero. If $Q'_{x}(b,a)=Q'_{y}(b,a)=0$ and $P_{n-1}(b,a)\neq 0$ , there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called aparabolic branch, even when it does not have any parabola that is a curvilinear asymptote. If $Q'_{x}(b,a)=Q'_{y}(b,a)=P_{n-1}(b,a)=0,$ the curve has a singular point at infinity which may have several asymptotes or parabolic branches.

Over the complex numbers,P_n splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals,P_n splits in factors that are linear or quadratic factors. 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. It may also occur that such a multiple linear factor corresponds to two complex conjugate branches, and does not corresponds to any infinite branch of the real curve. For example, the curvex⁴ +y² - 1 = 0 has no real points outside the square $|x|\leq 1,|y|\leq 1$ , but its highest order term gives the linear factorx with multiplicity 4, leading to the unique asymptotex=0.

Asymptotic cone

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Hyperbolas, obtained cutting the same right circular cone with a plane and their asymptotes

Thehyperbola

{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1

has the two asymptotes

y=\pm {\frac {b}{a}}x.

The equation for the union of these two lines is

{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=0.

Similarly, thehyperboloid

{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1

is said to have theasymptotic cone^[16]^[17]

{\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=0.

The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.

More generally, consider a surface that has an implicit equation $P_{d}(x,y,z)+P_{d-2}(x,y,z)+\cdots P_{0}=0,$ where the $P_{i}$ arehomogeneous polynomials of degree $i {\displaystyle i}$ and $P_{d-1}=0$ . Then the equation $P_{d}(x,y,z)=0$ defines acone which is centered at the origin. It is called anasymptotic cone, because the distance to the cone of a point of the surface tends to zero when the point on the surface tends to infinity.

References

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General references

Kuptsov, L.P. (2001) [1994],"Asymptote",Encyclopedia of Mathematics,EMS Press

Specific references

^Williamson, Benjamin (1899),"Asymptotes",An elementary treatise on the differential calculus
^Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the Projective Plane",Mathematics Magazine,72 (3):183–192,CiteSeerX 10.1.1.502.72,doi:10.2307/2690881,JSTOR 2690881
^Oxford English Dictionary, second edition, 1989.
^D.E. Smith,History of Mathematics, vol 2 Dover (1958) p. 318
^Apostol, Tom M. (1967),Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), New York:John Wiley & Sons,ISBN 978-0-471-00005-1, §4.18.
^Reference for section:"Asymptote"The Penny Cyclopædia vol. 2, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London p. 541
^Pogorelov, A. V. (1959),Differential geometry, Translated from the first Russian ed. by L. F. Boron, Groningen: P. Noordhoff N. V.,MR 0114163, §8.
^Fowler, R. H. (1920),The elementary differential geometry of plane curves, Cambridge, University Press,hdl:2027/uc1.b4073882,ISBN 0-486-44277-2, p. 89ff.
^William Nicholson,The British enciclopaedia, or dictionary of arts and sciences; comprising an accurate and popular view of the present improved state of human knowledge, Vol. 5, 1809
^Frost, P.An elementary treatise on curve tracing (1918)online
^Fowler, R. H.The elementary differential geometry of plane curves Cambridge, University Press, 1920, pp 89ff.(online at archive.org)
^Frost, P.An elementary treatise on curve tracing, 1918, page 5
^C.G. Gibson (1998)Elementary Geometry of Algebraic Curves, § 12.6 Asymptotes,Cambridge University PressISBN 0-521-64140-3,
^Coolidge, Julian Lowell (1959),A treatise on algebraic plane curves, New York:Dover Publications,ISBN 0-486-49576-0,MR 0120551, pp. 40–44.
^Kunz, Ernst (2005),Introduction to plane algebraic curves, Boston, MA: Birkhäuser Boston,ISBN 978-0-8176-4381-2,MR 2156630, p. 121.
^L.P. Siceloff, G. Wentworth, D.E. SmithAnalytic geometry (1922) p. 271
^P. FrostSolid geometry (1875) This has a more general treatment of asymptotic surfaces.