nLab limit of a binary function approaching a diagonal

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Contents

Definition

Let FF be an Archimedean field and let IFI \subseteq F be an open interval in FF. Let us define the subset Δ #(I)I×I\Delta_{\#}(I) \subseteq I \times I of pairs of elements apart from the diagonal as

Δ #(I){(x,y):I×I|0<|xy|}\Delta_{\#}(I) \coloneqq \{(x,y):I \times I \vert 0 \lt \vert x - y \vert \}

Let UU be a set such that Δ #(I)U\Delta_{\#}(I) \subseteq U and UI×IU \subseteq I \times I. As a result, for every element xIx \in I, there is an indexed set

U(x){yI|(x,y)}U(x) \coloneqq \{y \in I \vert (x, y)\}

Given a partial binary function q:UFq:U \to F, the currying of qq results in the indexed function

q(x):{yU(x)|(x,y)}Fq(x): \{y \in U(x) \vert (x,y)\} \to F

for every element xIx \in I

A function g:IFg:I \to F is a limit of qq approaching the diagonal if for all xSx \in S the limit of the dependent function q(x)q(x) approaching xx is g(x)g(x). We can define a predicate that the qq has a limit approaching the diagonal as

hasLimitApproachingDiagonal(q)g:IF.xIlim yxq(x)(y)=g(x)hasLimitApproachingDiagonal(q) \coloneqq \forall g \colon I \to F. \forall x \in I \lim_{y \to x} q(x)(y) = g(x)

The type of all functions in UFU \to F that have a limit approaching the diagonal is defined as

DiagLimFunc(I,F){qUF|hasLimitApproachingDiagonal(q)}DiagLimFunc(I, F) \coloneqq \{q \in U \to F \vert hasLimitApproachingDiagonal(q)\}

As a result, there exists a function

lim (x,y)(x,x)()(x,y):DiagLimFunc(I,F)(IF)\lim_{(x, y) \to (x, x)} (-)(x, y): DiagLimFunc(I, F) \to (I \to F)

which returns the limit of a partial binary function q:DiagLimFunc(I,F)q:DiagLimFunc(I, F) approaching the diagonal and thus satisfies the equation

lim yxq(x)(y)=lim (x,y)(x,x)q(x,y)\lim_{y \to x} q(x)(y) = \lim_{(x, y) \to (x, x)} q(x, y)

for all xIx \in I

Properties

In an Archimedean field FF, the algebraic limit theorems are satisfied:

Proposition

(Limits preserve the zero function)

The limit of a binary function approaching a diagonal preserves the zero function.

Proof

For all xIx \in I, the following is true:

lim (x,y)(x,x)0(x,y)=lim yx0(x)(y)\lim_{(x, y) \to (x, x)} 0(x, y) = \lim_{y \to x} 0(x)(y)
lim (x,y)(x,x)0(x,y)=0\lim_{(x, y) \to (x, x)} 0(x, y) = 0
lim (x,y)(x,x)0(x,y)=0(x,x)\lim_{(x, y) \to (x, x)} 0(x, y) = 0(x, x)

Thus, the limit of a binary function approaching a diagonal preserves the zero function.

Proposition

(Limits preserve addition of functions)

The limit of a binary function approaching a diagonal preserves addition of functions .

Proof

For all elements cIc \in I and functions f:IFf:I \to F and g:IFg:I \to F such that

lim xcf(x)=clim xcg(x)=c\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c

the following is true:

lim (x,y)(x,x)f(x,y)+g(x,y)=lim (x,y)(x,x)(f+g)(x,y)\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{(x, y) \to (x, x)} (f + g)(x, y)
lim (x,y)(x,x)f(x,y)+g(x,y)=lim yx(f+g)(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{y \to x} (f + g)(x)(y)
lim (x,y)(x,x)f(x,y)+g(x,y)=lim yxf(x)(y)+g(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{y \to x} f(x)(y) + g(x)(y)
lim (x,y)(x,x)f(x,y)+g(x,y)=lim yxf(x)(y)+lim yxg(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{y \to x} f(x)(y) + \lim_{y \to x} g(x)(y)
lim (x,y)(x,x)f(x,y)+g(x,y)=lim (x,y)(x,x)f(x,y)+lim (x,y)(x,x)g(x,y)\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{(x, y) \to (x, x)} f(x, y) + \lim_{(x, y) \to (x, x)} g(x, y)

Thus, the limit of a binary function approaching a diagonal preserves addition of functions .

Proposition

(Limits preserve negation of functions)

The limit of a binary function approaching a diagonal preserves negation of functions.

Proof

For all elements cIc \in I and functions f:IFf:I \to F such that

lim xcf(x)=c\lim_{x \to c} f(x) = c

the following is true:

lim (x,y)(x,x)f(x,y)=lim yxf(x)(y)\lim_{(x, y) \to (x, x)} -f(x, y) = \lim_{y \to x} -f(x)(y)
lim (x,y)(x,x)f(x,y)=lim yxf(x)(y)\lim_{(x, y) \to (x, x)} -f(x, y) = -\lim_{y \to x} f(x)(y)
lim (x,y)(x,x)f(x,y)=lim (x,y)(x,x)f(x,y)\lim_{(x, y) \to (x, x)} -f(x, y) = -\lim_{(x, y) \to (x, x)} f(x, y)

Thus, the limit of a binary function approaching a diagonal preserves negation of functions.

Proposition

(Limits preserve subtraction of functions)

The limit of a binary function approaching a diagonal preserves subtraction of functions .

Proof

For all elements cIc \in I and functions f:IFf:I \to F and g:IFg:I \to F such that

lim xcf(x)=clim xcg(x)=c\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c

the following is true:

lim (x,y)(x,x)f(x,y)g(x,y)=lim (x,y)(x,x)(fg)(x,y)\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{(x, y) \to (x, x)} (f - g)(x, y)
lim (x,y)(x,x)f(x,y)g(x,y)=lim yx(fg)(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{y \to x} (f - g)(x)(y)
lim (x,y)(x,x)f(x,y)g(x,y)=lim yxf(x)(y)g(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{y \to x} f(x)(y) - g(x)(y)
lim (x,y)(x,x)f(x,y)g(x,y)=lim yxf(x)(y)lim yxg(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{y \to x} f(x)(y) - \lim_{y \to x} g(x)(y)
lim (x,y)(x,x)f(x,y)g(x,y)=lim (x,y)(x,x)f(x,y)lim (x,y)(x,x)g(x,y)\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{(x, y) \to (x, x)} f(x, y) - \lim_{(x, y) \to (x, x)} g(x, y)

Thus, the limit of a binary function approaching a diagonal preserves addition of functions .

Proposition

(Limits preserve the left multiplicative \mathbb{Z}-action of functions)

The limit of a binary function approaching a diagonal preserves the left multiplicative \mathbb{Z}-action of functions.

Proof

For all elements cIc \in I and functions f:IFf:I \to F such that

lim xcf(x)=c\lim_{x \to c} f(x) = c

and integers nn \in \mathbb{Z}, the following is true:

lim (x,y)(x,x)nf(x,y)=lim (x,y)(x,x)(nf)(x,y)\lim_{(x, y) \to (x, x)} n f(x, y) = \lim_{(x, y) \to (x, x)} (n f)(x, y)
lim (x,y)(x,x)nf(x,y)=lim yx(nf)(x)(y)\lim_{(x, y) \to (x, x)} n f(x, y) = \lim_{y \to x} (n f)(x)(y)
lim (x,y)(x,x)nf(x,y)=lim yxnf(x)(y)\lim_{(x, y) \to (x, x)} n f(x, y) = \lim_{y \to x} n f(x)(y)
lim (x,y)(x,x)nf(x,y)=nlim yxf(x)(y)\lim_{(x, y) \to (x, x)} n f(x, y) = n \lim_{y \to x} f(x)(y)
lim (x,y)(x,x)nf(x,y)=nlim (x,y)(x,x)f(x,y)\lim_{(x, y) \to (x, x)} n f(x, y) = n \lim_{(x, y) \to (x, x)} f(x, y)

Thus, the limit of a binary function approaching a diagonal preserves the left multiplicative \mathbb{Z}-action of functions.

Proposition

(Limits preserve left multiplication by scalars)

The limit of a binary function approaching a diagonal preserves left multiplication by scalars.

Proof

For all elements cIc \in I and functions f:IFf:I \to F such that

lim xcf(x)=c\lim_{x \to c} f(x) = c

and elements aFa \in F, the following is true:

lim (x,y)(x,x)af(x,y)=lim (x,y)(x,x)(af)(x,y)\lim_{(x, y) \to (x, x)} a f(x, y) = \lim_{(x, y) \to (x, x)} (a f)(x, y)
lim (x,y)(x,x)af(x,y)=lim yx(af)(x)(y)\lim_{(x, y) \to (x, x)} a f(x, y) = \lim_{y \to x} (a f)(x)(y)
lim (x,y)(x,x)af(x,y)=lim yxaf(x)(y)\lim_{(x, y) \to (x, x)} a f(x, y) = \lim_{y \to x} a f(x)(y)
lim (x,y)(x,x)af(x,y)=alim yxf(x)(y)\lim_{(x, y) \to (x, x)} a f(x, y) = a \lim_{y \to x} f(x)(y)
lim (x,y)(x,x)af(x,y)=alim (x,y)(x,x)f(x,y)\lim_{(x, y) \to (x, x)} a f(x, y) = a \lim_{(x, y) \to (x, x)} f(x, y)

Thus, the limit of a binary function approaching a diagonal preserves left multiplication by scalars.

Proposition

(Limits preserve the constant one function)

The limit of a binary function approaching a diagonal preserves the constant one function.

Proof

For all xIx \in I, the following is true:

lim (x,y)(x,x)1(x,y)=lim yx1(x)(y)\lim_{(x, y) \to (x, x)} 1(x, y) = \lim_{y \to x} 1(x)(y)
lim (x,y)(x,x)1(x,y)=1\lim_{(x, y) \to (x, x)} 1(x, y) = 1
lim (x,y)(x,x)1(x,y)=1(x,x)\lim_{(x, y) \to (x, x)} 1(x, y) = 1(x, x)

Thus, the limit of a binary function approaching a diagonal preserves the constant one function.

Proposition

(Limits preserve multiplication of functions)

The limit of a binary function approaching a diagonal preserves multiplication of functions .

Proof

For all elements cIc \in I and functions f:IFf:I \to F and g:IFg:I \to F such that

lim xcf(x)=clim xcg(x)=c\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c

the following is true:

lim (x,y)(x,x)f(x,y)g(x,y)=lim (x,y)(x,x)(fg)(x,y)\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{(x, y) \to (x, x)} (f \cdot g)(x, y)
lim (x,y)(x,x)f(x,y)g(x,y)=lim yx(fg)(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{y \to x} (f \cdot g)(x)(y)
lim (x,y)(x,x)f(x,y)g(x,y)=lim yxf(x)(y)g(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{y \to x} f(x)(y) \cdot g(x)(y)
lim (x,y)(x,x)f(x,y)g(x,y)=lim yxf(x)(y)lim yxg(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{y \to x} f(x)(y) \cdot \lim_{y \to x} g(x)(y)
lim (x,y)(x,x)f(x,y)g(x,y)=lim (x,y)(x,x)f(x,y)lim (x,y)(x,x)g(x,y)\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{(x, y) \to (x, x)} f(x, y) \cdot \lim_{(x, y) \to (x, x)} g(x, y)

Thus, the limit of a binary function approaching a diagonal preserves multiplication of functions .

Proposition

(Limits preserve the powers of functions)

The limit of a binary function approaching a diagonal preserves powers of functions.

Proof

For all elements cIc \in I and functions f:IFf:I \to F such that

lim xcf(x)=c\lim_{x \to c} f(x) = c

and natural number nn \in \mathbb{N}, the following is true:

lim (x,y)(x,x)f(x,y) n=lim (x,y)(x,x)(f n)(x,y)\lim_{(x, y) \to (x, x)} f(x, y)^n = \lim_{(x, y) \to (x, x)} (f^n)(x, y)
lim (x,y)(x,x)f(x,y) n=lim yx(f n)(x)(y)\lim_{(x, y) \to (x, x)} f(x, y)^n = \lim_{y \to x} (f^n)(x)(y)
lim (x,y)(x,x)f(x,y) n=lim yx(f(x)(y)) n\lim_{(x, y) \to (x, x)} f(x, y)^n = \lim_{y \to x} (f(x)(y))^n
lim (x,y)(x,x)f(x,y) n=(lim yxf(x)(y)) n\lim_{(x, y) \to (x, x)} f(x, y)^n = \left(\lim_{y \to x} f(x)(y)\right)^n
lim (x,y)(x,x)f(x,y) n=(lim (x,y)(x,x)f(x,y)) n\lim_{(x, y) \to (x, x)} f(x, y)^n = \left(\lim_{(x, y) \to (x, x)} f(x, y)\right)^n

Thus, the limit of a binary function approaching a diagonal preserves powers of functions.

Limits preserve reciprocals of functions

Proposition

(Limits preserve reciprocals of functions)

The limit of a binary function approaching a diagonal preserves reciprocals of functions.

Proof

Let x 1x^{-1} be another notation for 1x\frac{1}{x}. For all elements cIc \in I and functions f:IFf:I \to F such that

lim xcf(x)=clim (x,y)(x,x)f(x,y)#0\lim_{x \to c} f(x) = c \qquad \lim_{(x, y) \to (x, x)} f(x, y) \# 0

the following is true:

lim (x,y)(x,x)f(x,y) 1=lim (x,y)(x,x)(f 1)(x,y)\lim_{(x, y) \to (x, x)} f(x, y)^{-1} = \lim_{(x, y) \to (x, x)} (f^{-1})(x, y)
lim (x,y)(x,x)f(x,y) 1=lim yx(f 1)(x)(y)\lim_{(x, y) \to (x, x)} f(x, y)^{-1} = \lim_{y \to x} (f^{-1})(x)(y)
lim (x,y)(x,x)f(x,y) 1=lim yx(f(x)(y)) 1\lim_{(x, y) \to (x, x)} f(x, y)^{-1} = \lim_{y \to x} (f(x)(y))^{-1}
lim (x,y)(x,x)f(x,y) 1=(lim yxf(x)(y)) 1\lim_{(x, y) \to (x, x)} f(x, y)^{-1} = \left(\lim_{y \to x} f(x)(y)\right)^{-1}
lim (x,y)(x,x)f(x,y) 1=(lim (x,y)(x,x)f(x,y)) 1\lim_{(x, y) \to (x, x)} f(x, y)^{-1} = \left(\lim_{(x, y) \to (x, x)} f(x, y)\right)^{-1}

Thus, the limit of a binary function approaching a diagonal preserves reciprocals of functions.

Proposition

(Limits preserve that the reciprocals of functions are multiplicative inverses)

The limit of a binary function approaching a diagonal preserves that the reciprocals of functions are multiplicative inverses.

Proof

Let x 1x^{-1} be another notation for 1x\frac{1}{x}. For all elements cIc \in I and functions f:IFf:I \to F such that

lim xcf(x)=c|lim (x,y)(x,x)f(x,y)|>0\lim_{x \to c} f(x) = c \qquad \left| \lim_{(x, y) \to (x, x)} f(x, y) \right| \gt 0

the following is true:

Where x 1x^{-1} is another notation for 1x\frac{1}{x}

lim (x,y)(x,x)f(x,y)f(x,y) 1=lim (x,y)(x,x)(ff 1)(x,y)\lim_{(x, y) \to (x, x)} f(x, y) \cdot {f(x, y)}^{-1} = \lim_{(x, y) \to (x, x)} (f \cdot f^{-1})(x, y)
lim (x,y)(x,x)f(x,y)f(x,y) 1=lim yx(ff 1)(x)(y)\lim_{(x, y) \to (x, x)} f(x, y) \cdot {f(x, y)}^{-1} = \lim_{y \to x} (f \cdot f^{-1})(x)(y)
lim (x,y)(x,x)f(x,y)f(x,y) 1=lim yxf(x)(y)f(x)(y) 1\lim_{(x, y) \to (x, x)} f(x, y) \cdot {f(x, y)}^{-1} = \lim_{y \to x} f(x)(y) \cdot {f(x)(y)}^{-1}
lim (x,y)(x,x)f(x,y)f(x,y) 1=1\lim_{(x, y) \to (x, x)} f(x, y) \cdot {f(x, y)}^{-1} = 1

Thus, limit of a binary function approaching a diagonal preserves that the reciprocals of functions are multiplicative inverses.

See also

Last revised on June 3, 2022 at 17:24:12. See the history of this page for a list of all contributions to it.