limit,
n. 1. a value that is approached increasingly closely by a function f(x) or a sequence an as the independent variable, x or n, increases without restriction. If for any small ɛ > 0 there is a large integer N such that
|f(x) - k| < ɛ, for all x > N,
k is said to be the limit of f(x) as x tends to infinity, written
.
2a. a value that a function f(x) approaches increasingly closely as the independent variable, x, approaches a stated value, a. If for any ɛ > 0 there is a δ such that
| f(x) - k | < ɛ, for all x such that | x - a | < δ,
then k is the limit of f(x) as x tends to a, written
. If f(x) is continuous at a,
.
This formalism replaces the non-standard definition of limits in terms of infinitesimals: that f(x) is at most infinitesimally distant from f(a) for x infinitesimally distant from a.
b. a point at which the preceding condition holds either for values less than x, in which case it is a limit on the left (a left-hand limit), or for values greater than x, in which case it is a limit on the right (a right-hand limit). If a function has both a left-hand limit and a right-hand limit and they are equal, then it has a limit at the point.
3a. in a metric space, a value, k, such that, for any ɛ there is a δ such that
d(f(x), k) < ɛ, for all x such that d(x, a) < δ,
that is, as x tends to a. This is analogous to the preceding, with metrics replacing absolute value.
b. similarly, in a topological space, a value, k, such that, for any neighborhood N(k) of k, there is a neighborhood N(a) of a such that
f(x) ∈ N(k), for all x ∈ N(a),
that is, as x tends to a. 4. a point such that a sequence or net in a topological space is eventually in every neighborhood of the limiting point: that is, xα tends to x if for each neighborhood V of x there exists β in the directed set A such that xα belongs to V for α ≥ β. 5. (Measure theory) a measurable function, f, such that a given sequence { fn}, of measurable functions is convergent in measure to f; that is, for every ɛ > 0, there is an N such that
μ({ x : | fn(x) - f(x)| > ɛ}) > ɛ
for all n > N. See tend to, epsilon-delta notation, converge, continuous, sum (sense 2). See also limit inferior, limit superior.
|f(x) - k| < ɛ, for all x > N,
k is said to be the limit of f(x) as x tends to infinity, written
.| f(x) - k | < ɛ, for all x such that | x - a | < δ,
then k is the limit of f(x) as x tends to a, written
..b. a point at which the preceding condition holds either for values less than x, in which case it is a limit on the left (a left-hand limit), or for values greater than x, in which case it is a limit on the right (a right-hand limit). If a function has both a left-hand limit and a right-hand limit and they are equal, then it has a limit at the point.
3a. in a metric space, a value, k, such that, for any ɛ there is a δ such that
d(f(x), k) < ɛ, for all x such that d(x, a) < δ,
that is, as x tends to a. This is analogous to the preceding, with metrics replacing absolute value.
b. similarly, in a topological space, a value, k, such that, for any neighborhood N(k) of k, there is a neighborhood N(a) of a such that
f(x) ∈ N(k), for all x ∈ N(a),
that is, as x tends to a. 4. a point such that a sequence or net in a topological space is eventually in every neighborhood of the limiting point: that is, xα tends to x if for each neighborhood V of x there exists β in the directed set A such that xα belongs to V for α ≥ β. 5. (Measure theory) a measurable function, f, such that a given sequence { fn}, of measurable functions is convergent in measure to f; that is, for every ɛ > 0, there is an N such that
μ({ x : | fn(x) - f(x)| > ɛ}) > ɛ
for all n > N. See tend to, epsilon-delta notation, converge, continuous, sum (sense 2). See also limit inferior, limit superior.