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In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
One can state a one-sided comparison test by using limit superior. Let a n , b n ≥ 0 {\displaystyle a_{n},b_{n}\geq 0} for all n {\displaystyle n} . Then if lim sup n → ∞ a n b n = c {\displaystyle \limsup _{n\to \infty }{\frac {a_{n}}{b_{n}}}=c} with 0 ≤ c < ∞ {\displaystyle 0\leq c<\infty } and Σ n b n {\displaystyle \Sigma _{n}b ...
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.
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 ∞.
Hence, one can easily see that uniform convergence is a stronger property than pointwise convergence: the existence of uniform limit implies the existence and equality of pointwise limit: If x n , m → y m {\displaystyle x_{n,m}\to y_{m}} uniformly, then x n , m → y m {\displaystyle x_{n,m}\to y_{m}} pointwise.
() (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.