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  2. Limit inferior and limit superior - Wikipedia

    en.wikipedia.org/wiki/Limit_inferior_and_limit...

    That is, xlim sup X n if and only if there exists a subsequence (X n k) of (X n) such that xX 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, xlim inf X n if and only if there exists some m > 0 such that xX n for all n > m.

  3. List of limits - Wikipedia

    en.wikipedia.org/wiki/List_of_limits

    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 ...

  4. Limit (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Limit_(mathematics)

    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 Xx 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}.

  5. Interchange of limiting operations - Wikipedia

    en.wikipedia.org/wiki/Interchange_of_limiting...

    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 ∞.

  6. Limit of a function - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_function

    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.

  7. Set-theoretic limit - Wikipedia

    en.wikipedia.org/wiki/Set-theoretic_limit

    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 ...

  8. Characterizations of the exponential function - Wikipedia

    en.wikipedia.org/wiki/Characterizations_of_the...

    () (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 !

  9. Computation in the limit - Wikipedia

    en.wikipedia.org/wiki/Computation_in_the_limit

    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.