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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.
In these limits, the infinitesimal change is often denoted or .If () is differentiable at , (+) = ′ ().This is the definition of the derivative.All differentiation rules can also be reframed as rules involving limits.
On one hand, the limit as n approaches infinity of a sequence {a n} is simply the limit at infinity of a function a(n) —defined on the natural numbers {n}. 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 ...
For every sequence (x n) in X − {p} that converges to p, the sequence f(x n) converges to L. If L is the limit (in the sense above) of f as x approaches p, then it is a sequential limit as well, however the converse need not hold in general.
supremum = least upper bound. A lower bound of a subset of a partially ordered set (,) is an element of such that . for all .; A lower bound of is called an infimum (or greatest lower bound, or meet) of if
Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.
Let f 1, f 2, ... denote a sequence of real-valued measurable functions defined on a measure space (S,Σ,μ).If there exists a Lebesgue-integrable function g on S which dominates the sequence in absolute value, meaning that |f n | ≤ g for all natural numbers n, then all f n as well as the limit inferior and the limit superior of the f n are integrable and
This sequence converges uniformly on S to the zero function and the limit, 0, is reached in a finite number of steps: for every x ≥ 0, if n > x, then f n (x) = 0. However, every function f n has integral −1. Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0).