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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
The first three functions have points for which the limit does not exist, while the function = is not defined at =, but its limit does exist. respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. [7] If the one-sided limits exist at p, but are unequal, then there is no limit at ...
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist.
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 ...
The adjective indeterminate does not imply that the limit does not exist, as many of the examples above show. In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated.
The function in example 2, a jump discontinuity. Consider the function = {< = > Then, the point = is a jump discontinuity.. In this case, a single limit does not exist because the one-sided limits, and + exist and are finite, but are not equal: since, +, the limit does not exist.
In this case, a single limit does not exist because the one-sided limits, L − and L +, exist and are finite, but are not equal: since, L − ≠ L +, the limit L does not exist. Then, x 0 is called a jump discontinuity, step discontinuity, or discontinuity of the first kind.
If the preordered set (,) also happens to be a partially ordered set (or more generally, if the restriction (,) is a partially ordered set) then is a maximal element of if and only if contains no element strictly greater than ; explicitly, this means that there does not exist any element such that and .