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

Operators