Search results
Results From The WOW.Com Content Network
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.
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.
If () for all x in an interval that contains c, except possibly c itself, and the limit of () and () both exist at c, then [5] () If lim x → c f ( x ) = lim x → c h ( x ) = L {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L} and f ( x ) ≤ g ( x ) ≤ h ( x ) {\displaystyle f(x)\leq g(x)\leq h(x)} for all x in an open interval that ...
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
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 ...
In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form , = (,), (,) = ((,)),or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value ...
If X is the continuous dual space of some other Banach space Y, then X is said to have the weak-∗ Opial property if, whenever (x n) n∈N is a sequence in X converging weakly-∗ to some x 0 ∈ X and x ≠ x 0, it follows that
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).