Limit (mathematics)
In functions

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 ".

Formally, the definition of the "limit of f (x) {\displaystyle f(x)} as x {\displaystyle x} approaches c {\displaystyle c} " is given as follows. The limit is a real number L {\displaystyle L} so that, given an arbitrary real number ϵ > 0 {\displaystyle \epsilon >0} (thought of as the "error"), there is a δ > 0 {\displaystyle \delta >0} such that, for any x {\displaystyle x} satisfying 0 < | x − c | < δ {\displaystyle 0<|x-c|<\delta }, it holds that | f (x) − L | < ϵ {\displaystyle |f(x)-L|<\epsilon }. This is known as the (ε, δ)-definition of limit.

The inequality 0 < | x − c | {\displaystyle 0<|x-c|} is used to exclude c {\displaystyle c} from the set of points under consideration, but some authors do not include this in their definition of limits, replacing 0 < | x − c | < δ {\displaystyle 0<|x-c|<\delta } with simply | x − c | < δ {\displaystyle |x-c|<\delta }. This replacement is equivalent to additionally requiring that f {\displaystyle f} be continuous at c {\displaystyle c}.

It can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions. The equivalent definition is given as follows. First observe that for every sequence { x n } {\displaystyle \{x_{n}\}} in the domain of f {\displaystyle f}, there is an associated sequence { f (x n) } {\displaystyle \{f(x_{n})\}}, the image of the sequence under f {\displaystyle f}. The limit is a real number L {\displaystyle L} so that, for all sequences x n → c {\displaystyle x_{n}\rightarrow c}, the associated sequence f (x n) → L {\displaystyle f(x_{n})\rightarrow L}.
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