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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 these limits, the infinitesimal change is often denoted or .If () is differentiable at , (+) = ′ ().This is the definition of the derivative.All differentiation rules can also be reframed as rules involving limits.
This can also apply to limits: see Vanish at infinity. weak, weaker The converse of strong. well-defined Accurately and precisely described or specified. For example, sometimes a definition relies on a choice of some object; the result of the definition must then be independent of this choice.
Limit of a function (ε,_δ)-definition of limit, formal definition of the mathematical notion of limit; Limit of a sequence; One-sided limit, either of the two limits of a function as a specified point is approached from below or from above; Limit inferior and limit superior; Limit of a net; Limit point, in topological spaces; Limit (category ...
Limits describe the behavior of a function at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of the real number system (as a metric space with the least-upper-bound property). In this treatment, calculus is a collection of techniques for manipulating certain limits.
lim – limit of a sequence, or of a function. lim inf – limit inferior. lim sup – limit superior. LLN – law of large numbers. ln – natural logarithm, log e. lnp1 – natural logarithm plus 1 function. ln1p – natural logarithm plus 1 function. log – logarithm. (If without a subscript, this may mean either log 10 or log e.)
If the terms in the sequence are real numbers, the limit superior and limit inferior always exist, as the real numbers together with ±∞ (i.e. the extended real number line) are complete. More generally, these definitions make sense in any partially ordered set, provided the suprema and infima exist, such as in a complete lattice.
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups , rings , vector spaces or in general objects from any category .