Search results
Results from the WOW.Com Content Network
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
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 ...
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.
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]
In those limits, the number of equations usually decreases, their order reduces, nonlinear equations can be replaced by linear ones, the initial system becomes averaged in a certain sense, and so on. All these idealizations, different as they may seem, increase the degree of symmetry of the mathematical model of the phenomenon under consideration.
In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point. A function is continuous at a limit point p of and in its domain if and only if f ( p ) is the (or, in the general case, a ) limit of f ( x ) as x tends to p .
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.
Composed in 1669, [4] during the mid-part of that year probably, [5] from ideas Newton had acquired during the period 1665–1666. [4] Newton wrote And whatever the common Analysis performs by Means of Equations of a finite number of Terms (provided that can be done) this new method can always perform the same by means of infinite Equations.