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This is a list of limits for common functions such as elementary functions. In this article, the terms a , b and c are constants with respect to x . Limits for general functions
In computer programming, bounds checking is any method of detecting whether a variable is within some bounds before it is used. It is usually used to ensure that a number fits into a given type (range checking), or that a variable being used as an array index is within the bounds of the array (index checking).
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 ...
For such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q), excluding the two lines x = p and y = q. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals L, then the multiple limit exists and also equals L. The ...
Relation to limits of sequences If S ≠ ∅ {\displaystyle S\neq \varnothing } is any non-empty set of real numbers then there always exists a non-decreasing sequence s 1 ≤ s 2 ≤ ⋯ {\displaystyle s_{1}\leq s_{2}\leq \cdots } in S {\displaystyle S} such that lim n → ∞ s n = sup S . {\displaystyle \lim _{n\to \infty }s_{n}=\sup S.}
Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits. [14] There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests.
The Bekenstein bound limits the amount of information that can be stored within a spherical volume to the entropy of a black hole with the same surface area. Thermodynamics limit the data storage of a system based on its energy, number of particles and particle modes. In practice, it is a stronger bound than the Bekenstein bound.
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 ...