nLab limit of a binary function approaching a diagonal

Definition

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

$$\Delta_{\#}(I) \coloneqq \{(x,y):I \times I \vert 0 \lt \vert x - y \vert \}$$

Let $U$ be a set such that $\Delta_{\#}(I) \subseteq U$ and $U \subseteq I \times I$. As a result, for every element $x \in I$, there is an indexed set

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

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

$$q(x): \{y \in U(x) \vert (x,y)\} \to F$$

for every element $x \in I$

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

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

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

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

As a result, there exists a function

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

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

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

for all $x \in I$

Properties

In an Archimedean field $F$, 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 $x \in I$, the following is true:

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

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

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

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 $c \in I$ and functions $f:I \to F$ and $g:I \to F$ such that

$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$

the following is true:

$$\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{(x, y) \to (x, x)} (f + g)(x, y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{y \to x} (f + g)(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{y \to x} f(x)(y) + g(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{y \to x} f(x)(y) + \lim_{y \to x} g(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) + g(x, y) = \lim_{(x, y) \to (x, x)} f(x, y) + \lim_{(x, y) \to (x, x)} g(x, y)$$

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 $c \in I$ and functions $f:I \to F$ such that

$$\lim_{x \to c} f(x) = c$$

the following is true:

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

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

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

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 $c \in I$ and functions $f:I \to F$ and $g:I \to F$ such that

$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$

the following is true:

$$\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{(x, y) \to (x, x)} (f - g)(x, y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{y \to x} (f - g)(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{y \to x} f(x)(y) - g(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{y \to x} f(x)(y) - \lim_{y \to x} g(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) - g(x, y) = \lim_{(x, y) \to (x, x)} f(x, y) - \lim_{(x, y) \to (x, x)} g(x, y)$$

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 $c \in I$ and functions $f:I \to F$ such that

$$\lim_{x \to c} f(x) = c$$

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

$$\lim_{(x, y) \to (x, x)} n f(x, y) = \lim_{(x, y) \to (x, x)} (n f)(x, y)$$

$$\lim_{(x, y) \to (x, x)} n f(x, y) = \lim_{y \to x} (n f)(x)(y)$$

$$\lim_{(x, y) \to (x, x)} n f(x, y) = \lim_{y \to x} n f(x)(y)$$

$$\lim_{(x, y) \to (x, x)} n f(x, y) = n \lim_{y \to x} f(x)(y)$$

$$\lim_{(x, y) \to (x, x)} n f(x, y) = n \lim_{(x, y) \to (x, x)} f(x, y)$$

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 $c \in I$ and functions $f:I \to F$ such that

$$\lim_{x \to c} f(x) = c$$

and elements $a \in F$, the following is true:

$$\lim_{(x, y) \to (x, x)} a f(x, y) = \lim_{(x, y) \to (x, x)} (a f)(x, y)$$

$$\lim_{(x, y) \to (x, x)} a f(x, y) = \lim_{y \to x} (a f)(x)(y)$$

$$\lim_{(x, y) \to (x, x)} a f(x, y) = \lim_{y \to x} a f(x)(y)$$

$$\lim_{(x, y) \to (x, x)} a f(x, y) = a \lim_{y \to x} f(x)(y)$$

$$\lim_{(x, y) \to (x, x)} a f(x, y) = a \lim_{(x, y) \to (x, x)} f(x, y)$$

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 $x \in I$, the following is true:

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

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

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

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 $c \in I$ and functions $f:I \to F$ and $g:I \to F$ such that

$$\lim_{x \to c} f(x) = c \qquad \lim_{x \to c} g(x) = c$$

the following is true:

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{(x, y) \to (x, x)} (f \cdot g)(x, y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{y \to x} (f \cdot g)(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{y \to x} f(x)(y) \cdot g(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{y \to x} f(x)(y) \cdot \lim_{y \to x} g(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot g(x, y) = \lim_{(x, y) \to (x, x)} f(x, y) \cdot \lim_{(x, y) \to (x, x)} g(x, y)$$

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 $c \in I$ and functions $f:I \to F$ such that

$$\lim_{x \to c} f(x) = c$$

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

$$\lim_{(x, y) \to (x, x)} f(x, y)^n = \lim_{(x, y) \to (x, x)} (f^n)(x, y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y)^n = \lim_{y \to x} (f^n)(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y)^n = \lim_{y \to x} (f(x)(y))^n$$

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

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

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

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

the following is true:

$$\lim_{(x, y) \to (x, x)} f(x, y)^{-1} = \lim_{(x, y) \to (x, x)} (f^{-1})(x, y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y)^{-1} = \lim_{y \to x} (f^{-1})(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y)^{-1} = \lim_{y \to x} (f(x)(y))^{-1}$$

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

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

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

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

the following is true:

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

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot {f(x, y)}^{-1} = \lim_{(x, y) \to (x, x)} (f \cdot f^{-1})(x, y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot {f(x, y)}^{-1} = \lim_{y \to x} (f \cdot f^{-1})(x)(y)$$

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot {f(x, y)}^{-1} = \lim_{y \to x} f(x)(y) \cdot {f(x)(y)}^{-1}$$

$$\lim_{(x, y) \to (x, x)} f(x, y) \cdot {f(x, y)}^{-1} = 1$$

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

See also

• limit of a function

• differentiable function

• difference quotient

• Newton-Leibniz operator