Limit (mathematics)

In mathematics, a limit is 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, a limit is usually written as

and is read as "the limit of f of n as n approaches c equals L". Here "lim" indicates limit, and the fact that function f(n) approaches the limit L as n approaches c is represented by the right arrow (→), as in

Suppose $f$ is a real-valued function and $c$ is a real number. Intuitively speaking, the expression

means that $f (x)$ can be made to be as close to $L$ as desired by making $x$ sufficiently close to $c$ . In that case, the above equation can be read as "the limit of $f$ of $x$ , as $x$ approaches $c$ , is $L$ ".

Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit. The definition uses $ε$ (the lowercase Greek letter epsilon) to represent any small positive number, so that " $f (x)$ becomes arbitrarily close to $L$ " means that $f (x)$ eventually lies in the interval $(L - ε, L + ε)$ , which can also be written using the absolute value sign as $| f (x) - L | < ε$ . The phrase "as $x$ approaches $c$ " then indicates that we refer to values of $x$ whose distance from $c$ is less than some positive number $δ$ (the lower case Greek letter delta)—that is, values of $x$ within either $(c - δ, c)$ or $(c, c + δ)$ , which can be expressed with $0 < | x - c | < δ$ . The first inequality means that the distance between $x$ and $c$ is greater than $0$ and that $x \neq c$ , while the second indicates that $x$ is within distance $δ$ of $c$ .

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