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  2. Limit inferior and limit superior - Wikipedia

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

    lim sup Y n = lim inf Y n = lim Y n = {1} lim sup Z n = lim inf Z n = lim Z n = {0} In each of these four cases, the elements of the limiting sets are not elements of any of the sets from the original sequence. The Ω limit (i.e., limit set) of a solution to a dynamic system is the outer limit of solution trajectories of the system.

  3. 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 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. [12] One such sequence would be {x 0 + 1/n}.

  4. Limit of a function - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_function

    Suppose M and N are subsets of metric spaces A and B, respectively, and f : M → N is defined between M and N, with x ∈ M, p a limit point of M and L ∈ N. It is said that the limit of f as x approaches p is L and write = if the following property holds:

  5. Limit of a sequence - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_sequence

    The plot of a convergent sequence {a n} is shown in blue. Here, one can see that the sequence is converging to the limit 0 as n increases. In the real numbers, a number is the limit of the sequence (), if the numbers in the sequence become closer and closer to , and not to any other number.

  6. 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 ...

  7. Computation in the limit - Wikipedia

    en.wikipedia.org/wiki/Computation_in_the_limit

    In computability theory, a function is called limit computable if it is the limit of a uniformly computable sequence of functions. The terms computable in the limit, limit recursive and recursively approximable are also used.

  8. Limit of distributions - Wikipedia

    en.wikipedia.org/wiki/Limit_of_distributions

    Given a sequence of distributions , its limit is the distribution given by [] = []for each test function , provided that distribution exists.The existence of the limit means that (1) for each , the limit of the sequence of numbers [] exists and that (2) the linear functional defined by the above formula is continuous with respect to the topology on the space of test functions.

  9. Spaces of test functions and distributions - Wikipedia

    en.wikipedia.org/wiki/Spaces_of_test_functions...

    One benefit of defining the canonical LF topology as the direct limit of a direct system is that we may immediately use the universal property of direct limits. Another benefit is that we can use well-known results from category theory to deduce that the canonical LF topology is actually independent of the particular choice of the directed ...