# Fungrim entry: 26ea9f

Symbol: Limit $\lim_{x \to a} f(x)$ Limiting value
Limit(f(x), For(x, a), P(x)) rendered as $\lim_{x \to a,\,P(x)} f(x)$ represents the limiting value of $f(x)$ for every sequence of $x$ satisfying $P(x)$ and approaching the limit point $a$.
If the predicate $P(x)$ is omitted, the expression renders correctly to LaTeX, but this form should be avoided since it is ambiguous whether it denotes a sequence limit, real limit or complex limit (or some other kind of limit). It is better to use SequenceLimit, RealLimit, LeftLimit, RightLimit or ComplexLimit.
The limit is always a deleted limit. That is, the value of $f(a)$ does not need to be equal to the limit and does not even need to be defined.
The expression f(x) is not required to be defined for all $x$ satisfying $P(x)$. It only needs to be defined for all $x$ in some neighborhood of the limit point and also satisfying $P(x)$.
The expression For(x, a) declares x as a locally bound variable within the scope of the arguments to this operator.
Definitions:
Fungrim symbol Notation Short description
Limit$\lim_{x \to a} f(x)$ Limiting value
SequenceLimit$\lim_{n \to a} f(n)$ Limiting value of sequence
RealLimit$\lim_{x \to a} f(x)$ Limiting value, real variable
LeftLimit$\lim_{x \to {a}^{-}} f(x)$ Limiting value, from the left
RightLimit$\lim_{x \to {a}^{+}} f(x)$ Limiting value, from the right
ComplexLimit$\lim_{z \to a} f(z)$ Limiting value, complex variable
Source code for this entry:
Entry(ID("26ea9f"),
SymbolDefinition(Limit, Limit(f(x), For(x, a)), "Limiting value"),
Description(SourceForm(Limit(f(x), For(x, a), P(x))), "rendered as", Limit(f(x), For(x, a), P(x)), "represents the limiting value of", f(x), "for every sequence of", x, "satisfying", P(x), "and approaching the limit point", a, "."),
Description("If the predicate", P(x), "is omitted, the expression renders correctly to LaTeX, ", "but this form should be avoided since it is ambiguous whether it denotes a sequence limit, ", "real limit or complex limit (or some other kind of limit). It is better to use", SourceForm(SequenceLimit), ",", SourceForm(RealLimit), ",", SourceForm(LeftLimit), ",", SourceForm(RightLimit), "or", SourceForm(ComplexLimit), "."),
Description("The limit is always a deleted limit. That is, the value of", f(a), "does not need to be equal to the limit and does not even need to be defined."),
Description("The expression", SourceForm(f(x)), "is not required to be defined for all", x, "satisfying", P(x), ".", "It only needs to be defined for all", x, "in some neighborhood of the limit point and also satisfying", P(x), "."),
Description("The expression", SourceForm(For(x, a)), "declares", SourceForm(x), "as a locally bound variable within the scope of the arguments to this operator."))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC