Dictionary Definition
asymptote n : a straight line that is the
limiting value of a curve; can be considered as tangent at
infinity; "the asymptote of the curve"
User Contributed Dictionary
English
Noun
 A straight line which a curve approaches arbitrarily closely, but never reaches, as they go to infinity. The limit of the curve, its tangent "at infinity".
Derived terms
Translations
a straight line which a curve approaches
arbitrarily closely
French
Noun
asymptote fExtensive Definition
A locally
connected curve A is said to be an asymptote of the locally
connected curve B when the following is true:
 For any positive \epsilon, there exist unbounded connected subsets (pieces of the respective curves) A^\prime\subseteq A and B^\prime\subseteq B, such that for every point in A^\prime its distance to the nearest point in B^\prime is lower than \epsilon.
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.
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.
Essentially, a linear asymptote is a straight
line a line that a graph approaches, but does not become identical
to.
Asymptotes, graphs and definitions
Meaning
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.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.
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:
 f(x) − g(x) → 0.
 f(x) / g(x) → 1.
 f(x) / g(x) has a nonzero limit.
More formally, curves A and B are asymptotic if
and only if there exist continuous functions x_A, y_A, x_B,
y_B\colon[0,1)\to\mathbb, such that all of the following conditions
are all true:

 \forall _\ (x_A(t),y_A(t))\in A
 \forall _\ (x_B(t),y_B(t))\in B
 \lim_ x_A(t)=\pm\infty \mbox \lim_ y_A(t)=\pm\infty
 \lim_ (x_A(t)x_B(t))=0
 \lim_ (y_A(t)y_B(t))=0
Multiple asymptotes, intersection
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.
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
Suppose f is a function. Then the line y = a is a
horizontal asymptote for f if
 \lim_ f(x) = a \,\mbox \lim_ f(x) = a.
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
 \lim_ f(x) = a \,\mbox \lim_ f(x) = b
Another example would be ƒ(x)=1/(x2+1), which has
a horizontal asymptote at y=0, as can be seen by the limit
 \lim_\frac=0
Vertical asymptotes
The line x = a is a vertical asymptote of a curve y=f(x) if at least one of the following statements is true: \lim_ f(x)=\pm\infty
 \lim_ f(x)=\pm\infty
 \lim_ f(x)=\pm\infty
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
 f(x) = \begin \frac & \mbox x > 0, \\ 5 & \mbox x \le 0 \end
As \lim_ f(x) = \infty, f(x) has a vertical
asymptote at 0, even though f(0) = 5.
Another example is ƒ(x) = 1/(x1) which has a
vertical asymptote of x=1 as shown by the limit
 \lim_\frac=\infty
Oblique asymptotes
When a linear asymptote is not parallel to the x or yaxis, it is called either an oblique asymptote or equivalently a slant asymptote. The function f(x) is asymptotic to y = mx + b if\lim_ f(x)(mx+b) = 0 \, \mbox \lim_ f(x)(mx+b)
= 0
Note that y = mx + b is never a vertical
asymptote, but can be a horizontal asymptote if m=0 (in which case
it is not an oblique asymptote).
An example is ƒ(x)=(x21)/x which has an oblique
asymptote of y=x (m=1, b=0) as seen in the limit
 \lim_f(x)x
 =\lim_\fracx
 =\lim_(x1/x)x
 =\lim_1/x=0
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
and equals zero. For example, to find the oblique asymptote of
y=25(x3+2x2+3x+4)/(5x2+6x+7), one can evaluate the limit
 \lim_\frac(mx+b)
 = \lim_5x+4+\frac+\fracmxb
 = \lim_ (5xmx)+ (4b)=0, \mbox m=5, b=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 \lim_f(x)g(x)
 =\lim_\frac(x^2+2x+3)
 =\lim_x^2+2x+3+\frac(x^2+2x+3)
 =\lim_\frac=0
Also, (ex)/(2x+1) is asymptotic to (ex)/x because
of the limit
 \lim_f(x)/g(x)
 =\lim_\frac
 =\lim_\frac=\frac
However, ex is not asymptotic to (ex)/x because
of the limit
 \lim_f(x)/g(x)
 =\lim_\frac
 =\lim_x=\infty
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.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 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
 f(x)=\frac=\frac
Oblique asymptotes
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
 f(x)=\frac=x+\frac
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.
 \frac
 =\fracx\frac
 \approx\fracx, \mboxx.
Translations of known functions
If a known function has an asymptote (such as y=0 for f(x)=ex), 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(xh)+k
 If y=b is a horizontal asymptote of f(x), then y=b+h is a horizontal asymptote of f(xh)+k
For example, f(x)=ex1+2 has horizontal asymptote
y=0+2=2, and no vertical or oblique asymptotes.
See also
asymptote in Bulgarian: Асимптота
asymptote in Catalan: Asímptota
asymptote in Czech: Asymptota
asymptote in Danish: Asymptote
asymptote in German: Asymptote
asymptote in Spanish: Asíntota
asymptote in Esperanto: Asimptoto
asymptote in French: Asymptote
asymptote in Italian: Asintoto
asymptote in Hebrew: אסימפטוטה
asymptote in Lithuanian: Asimptotė
asymptote in Macedonian: Асимптота
asymptote in Dutch: Asymptoot
asymptote in Japanese: 漸近線
asymptote in Polish: Asymptota
asymptote in Portuguese: Assímptota
asymptote in Romanian: Asimptotă
asymptote in Russian: Асимптота
asymptote in Slovenian: Asimptota
asymptote in Serbian: Асимптота
asymptote in Finnish: Asymptootti
asymptote in Swedish: Asymptot
asymptote in Turkish: Sonuşmaz
asymptote in Ukrainian: Асимптота
asymptote in Chinese: 渐近线
Synonyms, Antonyms and Related Words
approach, bottleneck, collision course,
concentralization,
concentration,
concourse, concurrence, confluence, conflux, congress, convergence, converging, crossing, focalization, focus, funnel, hub, meeting, mutual approach,
narrowing gap, radius,
spokes, tangent