In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives and integrals.

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.

In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a.

Limit of a function

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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

Informally, a function f assigns an output f(x) to every input x. The function has a limit L at an input p if f(x) is "close" to L whenever x is "close" to p. In other words, f(x) becomes closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to each input sufficiently close to p, the result is an output value that is arbitrarily close to L. If the inputs "close" to p are taken to values that are very different, the limit is said to not exist.

Definitions

The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.

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.

In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a.

Limit of a function

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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.

Informally, a function f assigns an output f(x) to every input x. The function has a limit L at an input p if f(x) is "close" to L whenever x is "close" to p. In other words, f(x) becomes closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to each input sufficiently close to p, the result is an output value that is arbitrarily close to L. If the inputs "close" to p are taken to values that are very different, the limit is said to not exist.

Definitions

The following definitions (known as (ε, δ)-definitions) are the generally accepted ones for the limit of a function in various contexts.

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