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Home > Limit of a function

In mathematics, the limit of a function is a fundamental concept in mathematical analysis.

Rather informally, to say that a function f has a limit y when x tends to a value x₀ (or to the infinity), is to say that the values taken by the expression f(x) get close to y when x gets close to x₀ (or gets infinitely big). Formal definitions, first devised around the end of the 19th century, are given below.

See net (topology) for a generalization of the concept of limit.

1 History

See mathematical analysis.

2 Formal definition

2.1 Functions on metric spaces

Suppose f : (M,d_M) -> (N,d_N) is a map between two metric spaces, p∈M and L∈N. We say that "the limit of f(x) is L as x approaches p" and write

if and only if

for every ε > 0 there exists a δ > 0 such that for all x in M with 0 < d_M(x, p) < δ, we have d_N(f(x), L) < ε.

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