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# Asymptote

[as-im-toht]
An asymptote of a real-valued function $y=f\left(x\right)$ is a curve which describes the behavior of $f$ as either $x$ or $y$ goes to infinity.

In other words, as one moves along the graph of $f\left(x\right)$ in some direction, the distance between it and the asymptote 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.

A linear asymptote is essentially a straight line to which a graphed curve becomes closer and closer but does not become identical.''

## Asymptotes, graphs and definitions

### 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) $A^primesubseteq A$ and $B^primesubseteq B$, such that for every point in $A^prime$ its distance to the nearest point in $B^prime$ is lower than $epsilon$.

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.

More formally, curves $A$ and $B$ are asymptotic if and only if there exist continuous functions $x_A, y_A, x_B, y_Bcolon\left[0,1\right)tomathbb\left\{R\right\}$, such that all of the following conditions are all true:

* $forall _\left\{tin\left[0,1\right)\right\} \left(x_A\left(t\right),y_A\left(t\right)\right)in A$
* $forall _\left\{tin\left[0,1\right)\right\} \left(x_B\left(t\right),y_B\left(t\right)\right)in B$
* $lim_\left\{tto 1\right\} x_A\left(t\right)=pminfty mbox\left\{ or \right\} lim_\left\{tto 1\right\} y_A\left(t\right)=pminfty$
* $lim_\left\{tto 1\right\} \left(x_A\left(t\right)-x_B\left(t\right)\right)=0$
* $lim_\left\{tto 1\right\} \left(y_A\left(t\right)-y_B\left(t\right)\right)=0$

### 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.

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

$lim_\left\{x to infty\right\} f\left(x\right) = a ,mbox\left\{ or \right\} lim_\left\{x to -infty\right\} f\left(x\right) = a.$

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

$lim_\left\{x to infty\right\} f\left(x\right) = a ,mbox\left\{ and \right\} lim_\left\{x to -infty\right\} f\left(x\right) = b$
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

$lim_\left\{xto infty\right\}frac\left\{1\right\}\left\{x^2+1\right\}=0$

### 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. $lim_\left\{x to a\right\} f\left(x\right)=pminfty$
2. $lim_\left\{x to a^\left\{-\right\}\right\} f\left(x\right)=pminfty$
3. $lim_\left\{x to a^\left\{+\right\}\right\} f\left(x\right)=pminfty$

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

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

As $lim_\left\{x to 0^\left\{+\right\}\right\} f\left(x\right) = infty$, f(x) has a vertical asymptote at 0, even though $f\left(0\right) = 5$.

Another example is ƒ(x) = 1/(x-1) which has a vertical asymptote of x=1 as shown by the limit

$lim_\left\{xto 1^+\right\}frac\left\{1\right\}\left\{x-1\right\}=infty$

### Oblique asymptotes

When a linear asymptote is not parallel to the x- or y-axis, it is called either an oblique asymptote or equivalently a slant asymptote. The function f(x) is asymptotic to y = mx + b if

$lim_\left\{x to infty\right\} f\left(x\right)-\left(mx+b\right) = 0 , mbox\left\{ or \right\} lim_\left\{x to -infty\right\} f\left(x\right)-\left(mx+b\right) = 0$

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)=(x2-1)/x which has an oblique asymptote of y=x (m=1, b=0) as seen in the limit

$lim_\left\{xtoinfty\right\}f\left(x\right)-x$
$=lim_\left\{xtoinfty\right\}frac\left\{x^2-1\right\}\left\{x\right\}-x$
$=lim_\left\{xtoinfty\right\}\left(x-1/x\right)-x$
$=lim_\left\{xtoinfty\right\}-1/x=0$

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

$lim_\left\{xtoinfty\right\}frac\left\{25\left(x^3+2x^2+3x+4\right)\right\}\left\{5x^2+6x+7\right\}-\left(mx+b\right)$
$= lim_\left\{xtoinfty\right\}5x+4+frac\left\{16x\right\}\left\{5x^2+6x+7\right\}+frac\left\{72\right\}\left\{5x^2+6x+7\right\}-mx-b$
$= lim_\left\{xtoinfty\right\} \left(5x-mx\right)+ \left(4-b\right)=0, mbox\left\{ when \right\} m=5, b=4$
So the oblique asymptote is y=5x+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
$lim_\left\{xtoinfty\right\}f\left(x\right)-g\left(x\right)$
$=lim_\left\{xtoinfty\right\}frac\left\{x^3+2x^2+3x+4\right\}\left\{x\right\}-\left(x^2+2x+3\right)$
$=lim_\left\{xtoinfty\right\}x^2+2x+3+frac\left\{4\right\}\left\{x\right\}-\left(x^2+2x+3\right)$
$=lim_\left\{xtoinfty\right\}frac\left\{4\right\}\left\{x\right\}=0$

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

$lim_\left\{xtoinfty\right\}f\left(x\right)/g\left(x\right)$
$=lim_\left\{xtoinfty\right\}frac\left\{e^x/\left(2x+1\right)\right\}\left\{e^x/x\right\}$
$=lim_\left\{xtoinfty\right\}frac\left\{x\right\}\left\{2x+1\right\}=frac\left\{1\right\}\left\{2\right\}$

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

$lim_\left\{xtoinfty\right\}f\left(x\right)/g\left(x\right)$
$=lim_\left\{xtoinfty\right\}frac\left\{e^x\right\}\left\{e^x/x\right\}$
$=lim_\left\{xtoinfty\right\}x=infty$

## 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) Horizontal/oblique asymptotes Example, asymptote
<0 y=0 $frac\left\{1\right\}\left\{x^2+1\right\}, y=0$
0 y="ratio of leading coefficients" $frac\left\{2x^2+7\right\}\left\{3x^2+x+12\right\}, y=frac\left\{2\right\}\left\{3\right\}$
1 1 oblique $frac\left\{2x^3\right\}\left\{3x^2+1\right\}, y=frac\left\{2\right\}\left\{3\right\}x$
>1 None $frac\left\{2x^4\right\}\left\{3x^2+1\right\}, mbox\left\{none\right\}$

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

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

#### 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

$f\left(x\right)=frac\left\{x^2+x+1\right\}\left\{x+1\right\}=x+frac\left\{1\right\}\left\{x+1\right\}$
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, 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.

$frac\left\{2x^3\right\}\left\{3x^2+1\right\}$
$=frac\left\{2\right\}\left\{3\right\}x-frac\left\{2x\right\}\left\{9x^2+3\right\}$
$approxfrac\left\{2\right\}\left\{3\right\}x, mbox\left\{for large \right\}|x|.$

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

For example, f(x)=ex-1+2 has horizontal asymptote y=0+2=2, and no vertical or oblique asymptotes.