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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 mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. [1] Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point. A function is continuous at a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p. There is another type of limit of a function, namely the ...
A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.
The existence theorem for limits states that if a category C has equalizers and all products indexed by the classes Ob(J) and Hom(J), then C has all limits of shape J. [ 1 ] : §V.2 Thm.1 In this case, the limit of a diagram F : J → C can be constructed as the equalizer of the two morphisms [ 1 ] : §V.2 Thm.2
Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers , often including positive and negative infinity to form the extended real line .
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 ...
A limit of a sequence of points () in a topological space is a special case of a limit of a function: the domain is in the space {+}, with the induced topology of the affinely extended real number system, the range is , and the function argument tends to +, which in this space is a limit point of .