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Download as PDF; Printable version; ... This is a list of limits for common functions such as elementary functions. ... 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 ...
Download as PDF; Printable version; ... Limit inferior and limit superior; Limit of a function; Limit of a sequence; ... Tannery's theorem
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 ...
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
This theorem does not hold if uniform convergence is replaced by pointwise convergence. For example, let ƒ n : [0, 1] → R be the sequence of functions ƒ n (x) = x n. Then each function ƒ n is continuous, but the sequence converges pointwise to the discontinuous function ƒ that is zero on [0, 1) but has ƒ(1) = 1. Another example is shown ...
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 ...
It is valid to move the limit inside the exponential function because this function is continuous. Now the exponent x {\displaystyle x} has been "moved down". The limit lim x → 0 + x ⋅ ln x {\displaystyle \lim _{x\to 0^{+}}x\cdot \ln x} is of the indeterminate form 0 · ∞ dealt with in an example above: L'Hôpital may be used to ...