In other words, as one moves along the graph of 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.''
A locally connected curve A is said to be an asymptote of the locally connected curve B when the following is true:
There are many different cases that can be treated separately, such as linear asymptotes (below), although intuitively the two functions become arbitrarily close.
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:
More formally, curves and are asymptotic if and only if there exist continuous functions , such that all of the following conditions are all true:
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.
Suppose f is a function. Then the line y = a is a horizontal asymptote for f if
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
Another example would be ƒ(x)=1/(x2+1), which has a horizontal asymptote at y=0, as can be seen by the limit
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
As , f(x) has a vertical asymptote at 0, even though .
Another example is ƒ(x) = 1/(x-1) which has a vertical asymptote of x=1 as shown by the limit
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
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
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
Also, (ex)/(2x+1) is asymptotic to (ex)/x because of the limit
However, ex is not asymptotic to (ex)/x because of the limit
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.
|deg(numerator) - deg(denominator)||Horizontal/oblique asymptotes||Example, asymptote|
|0||y="ratio of leading coefficients"|
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
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
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.
For example, f(x)=ex-1+2 has horizontal asymptote y=0+2=2, and no vertical or oblique asymptotes.