Asymptote

For other uses, see Asymptote (disambiguation).

In geometry, an asymptote of a curve is a way of describing its behavior far away from the origin by comparing it to another curve. Specifically, the second curve is an asymptote of the first if distance between the two approaches 0 as the points being considered tend to infinity. Informally, this means that the first curve gets closer to the second as it gets farther from the origin. An important case is when the asymptote is a straight line; this is called a linear asymptote (or simply asymptote if there is no chance of confusion).

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.

Asymptotes, graphs and definitions

Definition

$f(x)={\tfrac {1}{x}}$ graphed on Cartesian coordinates. The x and y axes are the asymptotes.

Let A be a curve defined parametrically by x = x(t), y = y(t). Say A goes to infinity at t=t₀ if either x(t) or y(t) goes to ±∞ as t approaches t₀, where t₀ may itself be ±∞. In this case, a curve B is said to be an asymptote of A if the distance between (x(t), y(t)) and B approaches 0 as t approaches t₀.

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). First, x → ∞ as t → ∞ and the distance from the curve to the x-axis is 1/t which approaches 0 as t → ∞. Therefore the x-axis is an asymptote of the curve. Also, y → ∞ as t → 0 from the right, and the distance between the curve and the y-axis is t which approaches 0 as t → 0. So the y-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.

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:

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

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.

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

$\lim _{x\to \infty }f(x)=a\,{\mbox{ or }}\lim _{x\to -\infty }f(x)=a.$

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

$\lim _{x\to \infty }f(x)=a\,{\mbox{ and }}\lim _{x\to -\infty }f(x)=b$

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

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

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:

1. $\lim _{x\to a}f(x)=\pm \infty$
2. $\lim _{x\to a^{-}}f(x)=\pm \infty$
3. $\lim _{x\to a^{+}}f(x)=\pm \infty .$

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

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

As $\lim _{x\to 0^{+}}f(x)=\infty$, 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

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

Oblique asymptotes

In the graph of $f(x)=x+{\tfrac {1}{x}}$, 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. A function f(x) is asymptotic to y = mx + c if

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

Note that y = mx + c 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)=(x²-1)/x which has an oblique asymptote of y=x (m=1, c=0) as seen in the limit

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

$=\lim _{x\to \infty }\left[{\frac {x^{2}-1}{x}}-x\right]$

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

$=\lim _{x\to \infty }-{\frac {1}{x}}=0$

Computationally identifying an oblique asymptote can be more difficult than a horizontal or vertical asymptote, in particular because the m and c might not be known. It is typical to evaluate the appropriate limit and choose m, c 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

$\lim _{x\to \infty }\left[{\frac {25(x^{3}+2x^{2}+3x+4)}{5x^{2}+6x+7}}-(mx+c)\right]$

$=\lim _{x\to \infty }\left[5x+4+{\frac {16x}{5x^{2}+6x+7}}+{\frac {72}{5x^{2}+6x+7}}-mx-c\right]$

$=\lim _{x\to \infty }\left[(5x-mx)+(4-c)\right]=0,{\mbox{ when }}m=5,c=4$

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

Nonlinear asymptotes

(x³+2x²+3x+4)/x asymptotic to x²+2x+3

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

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

$=\lim _{x\to \infty }\left[{\frac {x^{3}+2x^{2}+3x+4}{x}}-(x^{2}+2x+3)\right]$

$=\lim _{x\to \infty }\left[x^{2}+2x+3+{\frac {4}{x}}-(x^{2}+2x+3)\right]$

$=\lim _{x\to \infty }{\frac {4}{x}}=0$

(e^x)/(2x+1) asymptotic to (e^x)/x

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

$\lim _{x\to \infty }f(x)/g(x)$

$=\lim _{x\to \infty }{\frac {e^{x}/(2x+1)}{e^{x}/x}}$

$=\lim _{x\to \infty }{\frac {x}{2x+1}}={\frac {1}{2}}$

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

$\lim _{x\to \infty }f(x)/g(x)$

$=\lim _{x\to \infty }{\frac {e^{x}}{e^{x}/x}}$

$=\lim _{x\to \infty }x=\infty$

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.

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

deg(numerator) − deg(denominator)	asymptotes	example asymptote
< 0	y = 0	${\frac {1}{x^{2}+1}},y=0$
= 0	y = the ratio of leading coefficients	${\frac {2x^{2}+7}{3x^{2}+x+12}},y={\frac {2}{3}}$
= 1	1 oblique	${\frac {2x^{3}}{3x^{2}+1}},y={\frac {2}{3}}x$
> 1	none	${\frac {2x^{4}}{3x^{2}+1}},{\mbox{none}}$

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

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

Oblique asymptotes

Black: the graph of $f(x)={\frac {x^{2}+x+1}{x+1}}$. 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 after dividing 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 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, 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.

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

References

• Fowler, R. H. The elementary differential geometry of plane curves Cambridge, University Press, 1920, pp 89ff.

(online at archive.org)