In
analytic geometry, an asymptote () of a
curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates
tends to infinity. In
projective geometry and related contexts, an asymptote of a curve is a line which is
tangent to the curve at a
point at infinity
In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
.
The word asymptote is derived from the
Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ
priv. + σύν "together" + πτωτ-ός "fallen". 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.
There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the
graph of a
function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to 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 as ''x'' tends to
More generally, one curve is a ''curvilinear asymptote'' of another (as opposed to a ''linear asymptote'') if the distance between the two curves tends to zero as they tend to infinity, although the term ''asymptote'' by itself is usually reserved for linear asymptotes.
Asymptotes convey information about the behavior of curves ''in the large'', and determining the asymptotes of a function is an important step in sketching its graph. The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of
asymptotic analysis.
Introduction
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 (see
Line). Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.
Consider the graph of the function
shown in this section. The coordinates of the points on the curve are of the form
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
become larger and larger, say 100, 1,000, 10,000 ..., putting them far to the right of the illustration, the corresponding values of
, .01, .001, .0001, ..., become infinitesimal relative to the scale shown. But no matter how large
becomes, its reciprocal
is never 0, so the curve never actually touches the ''x''-axis. Similarly, as the values of
become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of
, 100, 1,000, 10,000 ..., become larger and larger. So the curve extends farther and farther upward as it comes closer and closer to the ''y''-axis. Thus, both the ''x'' and ''y''-axis are asymptotes of the curve. These ideas are part of the basis of concept of a
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
in mathematics, and this connection is explained more fully below.
Asymptotes of functions
The asymptotes most commonly encountered in the study of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
are of curves of the form . These can be computed using
limits and classified into ''horizontal'', ''vertical'' and ''oblique'' asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to +∞ or −∞. As the name indicates they are parallel to the ''x''-axis. Vertical asymptotes are vertical lines (perpendicular to the ''x''-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 as ''x'' tends to +∞ or −∞.
Vertical asymptotes
The line ''x'' = ''a'' is a ''vertical asymptote'' of the graph of the function if at least one of the following statements is true:
#
#
where
is the limit as ''x'' approaches the value ''a'' from the left (from lesser values), and
is the limit as ''x'' approaches ''a'' from the right.
For example, if ƒ(''x'') = ''x''/(''x''–1), the numerator approaches 1 and the denominator approaches 0 as ''x'' approaches 1. So
:
:
and the curve has a vertical asymptote ''x'' = 1.
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. For example, for the function
:
has a limit of +∞ as , ''ƒ''(''x'') has the vertical asymptote , even though ''ƒ''(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 (or
a vertical line in general) in more than one point. Moreover, if a function is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
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
:
at
.
This function has a vertical asymptote at
because
:
and
:
.
The derivative of
is the function
:
.
For the sequence of points
:
for
that approaches
both from the left and from the right, the values
are constantly
. Therefore, both
one-sided limits of
at
can be neither
nor
. Hence
doesn't have a vertical asymptote at
.
Horizontal asymptotes
''Horizontal asymptotes'' are horizontal lines that the graph of the function approaches as . The horizontal line ''y'' = ''c'' is a horizontal asymptote of the function ''y'' = ''ƒ''(''x'') if
:
or
.
In the first case, ''ƒ''(''x'') has ''y'' = ''c'' as asymptote when ''x'' tends to , and in the second ''ƒ''(''x'') has ''y'' = ''c'' as an asymptote as ''x'' tends to .
For example, the
arctangent function satisfies
:
and
So the line is a horizontal asymptote for the arctangent when ''x'' tends to , and is a horizontal asymptote for the arctangent when ''x'' 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 function has a horizontal asymptote at ''y'' = 0 when ''x'' tends both to and because, respectively,
:
Other common functions that have one or two horizontal asymptotes include (that has an
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
as it graph), the
Gaussian function the
error function, and the
logistic function.
Oblique asymptotes
When a linear asymptote is not parallel to the ''x''- or ''y''-axis, it is called an ''oblique asymptote'' or ''slant asymptote''. A function ''ƒ''(''x'') is asymptotic to the straight line (''m'' ≠ 0) if
:
In the first case the line is an oblique asymptote of ''ƒ''(''x'') when ''x'' tends to +∞, and in the second case the line is an oblique asymptote of ''ƒ''(''x'') when ''x'' tends to −∞.
An example is ''ƒ''(''x'') = ''x'' + 1/''x'', which has the oblique asymptote ''y'' = ''x'' (that is ''m'' = 1, ''n'' = 0) as seen in the limits
: