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If f and g are real-valued (or complex-valued) functions, then taking the limit of an operation on f(x) and g(x) (e.g., f + g, f − g, f × g, f / g, f g) under certain conditions is compatible with the operation of limits of f(x) and g(x). This fact is often called the algebraic limit theorem. The main condition needed to apply the following ...
This is a list of limits for common functions such as elementary functions. In this article, the terms a, ... This is known as the squeeze theorem. [1] [2] ...
Limit of a function. One-sided limit: either of the two limits of functions of a real variable x, as x approaches a point from above or below; List of limits: list of limits for common functions; Squeeze theorem: finds a limit of a function via comparison with two other functions; Limit superior and limit inferior; Modes of convergence. An ...
Bohr–Mollerup theorem (gamma function) Bohr–van Leeuwen theorem ; Bolyai–Gerwien theorem (discrete geometry) Bolzano's theorem (real analysis, calculus) Bolzano–Weierstrass theorem (real analysis, calculus) Bombieri's theorem (number theory) Bombieri–Friedlander–Iwaniec theorem (number theory) Bondareva–Shapley theorem
While this is often shown using the mean value theorem for real-valued functions, the same method can be applied for higher-dimensional functions by using the mean value inequality instead. Interchange of partial derivatives: Schwarz's theorem; Interchange of integrals: Fubini's theorem; Interchange of limit and integral: Dominated convergence ...
In calculus, the squeeze theorem (also known as the sandwich theorem, among other names [a]) is a theorem regarding the limit of a function that is bounded between two other functions. The squeeze theorem is used in calculus and mathematical analysis , typically to confirm the limit of a function via comparison with two other functions whose ...
The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function. More precisely, this theorem states that the uniform limit of uniformly continuous functions is uniformly continuous; for a locally compact space, continuity is equivalent to local ...
The uniform limit theorem also holds if continuity is replaced by uniform continuity. That is, if X and Y are metric spaces and ƒ n : X → Y is a sequence of uniformly continuous functions converging uniformly to a function ƒ, then ƒ must be uniformly continuous.