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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.
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}.
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.
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 ∞.
Here, one can see that the sequence is converging to the limit 0 as n increases. In the real numbers , a number L {\displaystyle L} is the limit of the sequence ( x n ) {\displaystyle (x_{n})} , if the numbers in the sequence become closer and closer to L {\displaystyle L} , and not to any other number.
The function () = + (), where denotes the sign function, has a left limit of , a right limit of +, and a function value of at the point =. In calculus, a one-sided limit refers to either one of the two limits of a function of a real variable as approaches a specified point either from the left or from the right.
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.
() (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 !