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In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.
If f(n) = n 2 + 3n, then as n becomes very large, the term 3n becomes insignificant compared to n 2. The function f(n) is said to be "asymptotically equivalent to n 2, as n → ∞". This is often written symbolically as f (n) ~ n 2, which is read as "f(n) is asymptotic to n 2". An example of an important asymptotic result is the prime number ...
The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x) A curve intersecting an asymptote infinitely many times In analytic geometry , an asymptote ( / ˈ æ s ɪ m p t oʊ t / ) of a curve is a line such that the distance between the curve and the line approaches zero as one or ...
An infinite discontinuity is the special case when either the left hand or right hand limit does not exist, specifically because it is infinite, and the other limit is either also infinite, or is some well defined finite number. In other words, the function has an infinite discontinuity when its graph has a vertical asymptote.
On one hand, the limit as n approaches infinity of a sequence {a n} is simply the limit at infinity of a function a(n) —defined on the natural numbers {n}. On the other hand, if X is the domain of a function f ( x ) and if the limit as n approaches infinity of f ( x n ) is L for every arbitrary sequence of points { x n } in X − x 0 which ...
Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, [1] Edmund Landau, [2] and others, collectively called Bachmann–Landau notation or asymptotic notation.
Regarding Backward Compositions Theorem, the example f 2n (z) = 1/2 and f 2n−1 (z) = −1/2 for S = {z : |z| < 1} demonstrates the inadequacy of simply requiring contraction into a compact subset, like Forward Compositions Theorem. For functions not necessarily analytic the Lipschitz condition suffices:
There are three basic rules for evaluating limits at infinity for a rational function = () (where p and q are polynomials): If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;