Search results
Results From The WOW.Com Content Network
That is, x ∈ lim sup X n if and only if there exists a subsequence (X n k) of (X n) such that x ∈ X n k for all k. lim inf X n consists of elements of X which belong to X n for all except finitely many n (i.e., for cofinitely many n). That is, x ∈ lim inf X n if and only if there exists some m > 0 such that x ∈ X n for all n > m.
x 0 can be any arbitrary real number. Sums. In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its ...
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 converges to x 0, then the limit of the function f(x) as x approaches x 0 is equal to L. [11] One such sequence would be {x 0 + 1/n}.
Examples abound, one of the simplest being that for a double sequence a m,n: it is not necessarily the case that the operations of taking the limits as m → ∞ and as n → ∞ can be freely interchanged. [4] For example take a m,n = 2 m − n. in which taking the limit first with respect to n gives 0, and with respect to m gives ∞.
Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, the output value can be made arbitrarily close to L if the input to f is taken sufficiently close to p.
In mathematics, the limit of a sequence of sets,, … (subsets of a common set ) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves ...
() (using x ≥ 0 to obtain the final inequality) so that: = One must use lim sup because it is not known if t n converges. For the other inequality, by the above expression for t n , if 2 ≤ m ≤ n , we have: 1 + x + x 2 2 !
One can think of limit computable functions as those admitting an eventually correct computable guessing procedure at their true value. A set is limit computable just when its characteristic function is limit computable. If the sequence is uniformly computable relative to D, then the function is limit computable in D.