The Epsilon-Delta Definition of a Limit

limit极限理论: 建立在 R完备性公理化的实数集: Set theory集合论之上的
即: limit 极限 是 x, y 的 Macro/Micro、动态变化/静态关系、无限/有限、量变/质变、过程/结果，绝对/相对, 任意(不确定性)/规律(确定性)，的统一
limit数量化(∀ϵ∃δ的Karl Weierstrass的Quantitative语言)理论:
lim x→c⁡ f⁢(x) = L: ∀ ϵ>0, ∃ δ>0 S.T. for all x≠c, if |x-c|<δ, then |f⁢(x)-L|<ϵ .

With respect to the great mathematician Karl Weierstrass who invented the rigorous formal Epsilon-Delta lang.:
0. static -> dynamic;

1. feeling -> qualitative comparative differences;
   absolute -> relative adjustable and measurable quantitative tolerance(differences) error.
   tolerances: | expected value - active value | or | theoretical value - empirical value |

2. Macro -> Micro, whole -> part,
   micro quantitative free variable: Epsilon-Delta. provide evidences for conclusion/validation of a theorem.
   since, first a theory MUST be the unity of macro and micro, whole and part,…, conforming the dialectics;
   second, most theory begin with phenomenons, hypothesis, ...
   and especially theorems in mathematics before publicly consensus,
   should be proved with rigorous formal conclusion and validation,
   which MUST based on other theorems,
   and micro quantitative free variable: Epsilon-Delta.

3. any <-> one
   existence and inverse.

4. The traditional notation for:
   δ(delta): the x-tolerance is denoted by the lowercase Greek letter delta, or δ ,
   ϵ(epsilon): the y-tolerance is denoted by the lowercase Greek letter epsilon, or ϵ.

5. If x is within δ(delta) units of c,
   then the corresponding value of y is within ϵ(epsilon) units of L.

6. The point is that δ and ϵ, being tolerances,
   can be any positive (but typically small) values .

7. given any ϵ>0, there exists δ>0,
   such that for all x≠c, if |x-c|<δ, then |f⁢(x)-L|<ϵ.

1.2 Epsilon-Delta Definition of a Limit

This section introduces the formal definition of a limit.
Many refer to this as “the epsilon-delta,” definition, referring to the letters ϵ and δ of the Greek alphabet.

Before we give the actual definition, let’s consider a few informal ways of describing a limit. Given a function y=f⁢(x) and an x-value, c, we say that “the limit of the function f, as x approaches c, is a value L”:

1. if “y tends to L” as “x tends to c.”
2. if “y approaches L” as “x approaches c.”
3. if “y is near L” whenever “x is near c.”

The problem with these definitions is that the words “tends,” “approach,” "near",
and especially “near” are not exact.
In what way does the variable x tend to, or approach, c?
How near do x and y have to be to c and L, respectively?

The definition we describe in this section comes from formalizing 3. A quick restatement gets us closer to what we want:

𝟑′. If x is within a certain tolerance level of c,
then the corresponding value y=f⁢(x) is within a certain tolerance level of L.

The traditional notation for:
the x-tolerance is the lowercase Greek letter delta, or δ ,
and the y-tolerance is denoted by lowercase epsilon, or ϵ .
One more rephrasing of 𝟑′ nearly gets us to the actual definition:

𝟑′′. If x is within δ units of c, then the corresponding value of y is within ϵ units of L.
We can write “x is within δ units of c” mathematically as
|x-c|<δ, which is equivalent to  c-δ<x<c+δ.

Letting the symbol “⟶” represent the word “implies,”
we can rewrite 𝟑′′ as
|x-c|<δ⟶|y-L|<ϵ or c-δ<x<c+δ⟶L-ϵ<y<L+ϵ.

The point is that δ and ϵ, being tolerances, can be any positive (but typically small) values .
Finally, we have the formal definition of the limit with the notation seen in the previous section.

Definition 1.2.1 The Limit of a Function
Let I be an open interval containing c , and let f be a function defined on I , except possibly at c . The limit of f(x), as x approaches c, is L , denoted by
lim x→c⁡ f⁢(x) = L,

means that given any ϵ>0, there exists δ>0 such that for all x≠c, if |x-c|<δ, then |f⁢(x)-L|<ϵ .

by courtesy of the author:
APEX Calculus I/II/III
University of North Dakota
Adapted from APEX Calculus
by Gregory Hartman, Ph.D., Department of Applied Mathematics, Virginia Military Institute
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With respect to the great mathematician Karl Weierstrass who invented the rigorous formal Epsilon-Delta lang.