**Method 0**: Wrong !!!

**Method 1** : Rigorous!

1) Prove [A] Convergent ?

2) If converge, find Limit ?

**Method 2**: by RMI Homomorphism

**Method 0**: Wrong !!!

**Method 1** : Rigorous!

1) Prove [A] Convergent ?

2) If converge, find Limit ?

**Method 2**: by RMI Homomorphism

**Philosophical Interpretation**:

when n tends to infinity, the ‘geometric mean’ of these event probabilities = , , …, , equals

**Method:**

**Notes**:

1. Geometric Mean:

2. See a similar limit question, but by the more theoretical rigorous French pedagogy :

1.2 Introduction to Limit

Analogy : Product to Cones (Limit)

2.1 Five categories used to define Limit:

- Index category (I)
- Category C: Functors (constant , D)
- Cones (Lim D)
- Functor Category [I, C ]:objects are (constant, D ), morphisms are natural transormations
- Set category of Hom-Set Cones [I,C] to Hom-Set C (c , Lim D )

2.2 Naturality

3.1 Examples: Equalizer

**CoLimit** = duality of Limit (inverted cone = co-Cone)

Functor **Continuity** = preserve Limit

such that

The above *scary* ‘epsilon-delta’ definition of “Limit” by the French mathematician Cauchy in 19th century is the standard **rigorous** definition in today’s **Analysis** textbooks.

It was not taught in my Cambridge GCE A-Level Pure Math in 1970s (still true today), but every French Baccalaureate Math student (Terminale, equivalent to JC 2 or Pre-U 2) knows it by heart. A Cornell University Math Dean recalled how he was told by his high-school teacher to memorise it — even though he did not fully understand — the “epsilon-delta” definition by “chanting”:

“for all epsilon, there is a delta ….”

(French: *Quelque soit epsilon, il existe un delta …*)

In this video, I am glad someone like Prof N. Wildberger recognised its “flaws” albeit rigorous, by suggesting another more intuitive definition:

◇ Cauchy’s “flaw”: ambiguous

Finding a certain

too counter-intuitive to grasp the idea by most university math students.

◇ **Intuitive Alternative**:

Find **any** 2 natural numbers m (“**Start**“), k (“**Scale**“) such that:

**A Simple Analogy in Life**:

Let P(n) = Any Person’s lifespan of age n

m = ‘Start’ Age to retire, say 60

k = ‘Scale’ of interval (in years, eg. 1 year)

A = Limit of a person’s lifespan, say 80 (male) or 85 (female)

As we grow older (n increases), from a certain “Start” point (m), our lifespan P(n) approaches the limit A, plus or minus k/n (years).

**Mathematical Rigour**:

**“Domain of Definition**” MUST first be considered prior to tackling :

1) Continuity 连续性

2) Differentiability 可微性

3) Integrability 可积性

4) Limit 极限

Mathematics is linked to Philiosophy! In this **life** (Domain of Definition ) we have a **limit** of lifespan (120 years = 2 x 60 years = 2个甲子).

In this same “**Domain of Definition**” our life is **Continuous** unless interrupted by unforseen circumstances (accident, diseases, war, …). At certain junctures of life we **Differentiate** ourselves by having some ‘smooth’ (not abrupt “V- “or “W-” shape) turns of event (eg. graduation from schools and university, National Service in military, marriage, children, jobs, honours/promotions, as well as failures …). It is only in this life you can **Integrate** these fruits of labor. Beyond this “Domain of Definition” life is meaniningless because we shall return to soil with nothing ….

See : **C.I.D .Triangle**

**Definition**:

has limit **a**

What if we *reverse* the order of the definition like this:

**∃ N** such that ∀ε > 0, ∀n ≥ N,

This means:

**Example**:

Proof:

Let’s prove it.

Therefore,

[**QED**]

[*Source*]: Excellent Introduction in Modern Math:

A Concise Introduction to Pure Mathematics, Third Edition (Chapman Hall/CRC Mathematics Series)

**Analysis** is the study of Functions, using the main tool “**Limit**“.

Limit problems appear in:

1. Continuity

2. Derivative

3. Integral

4. Sequence

**Multi-variable Functions **have different approach of Limit compared to Single-value functions.

Eg. L’Hôpital Rule is **not** applicable to Multi-variable Functions.

**Case 1:** Find the Limit (**L**) of

Solution:

Consider the point P(x,y) on f(x,y)

but when P moves along y=x straight line approaching (0,0),

ie. x->0, y=x->0,

From (1),(2),(3) there are 3 limits {0, 0, 1/2}, hence the Limit **L** does not exist.

**Case 2:**

Find limit L of

**Solution**:

Let y= kx

When x -> ,

f(x,y) independent of k => possible limit of 0

Prove:

Note:

[QED]

**Source:** Prof Zhang ShiZao 章士藻(1940-) “Collected Works of Mathematics Education” 数学教育文集

for Ecole Normale Supérieure in China 高等师范学院

[Video in French ] http://touch.dailymotion.com/video/x89qux

1. Basic:

|y|= 0 or > 0 for all y

2. **Limit**: ; x≠a

|x-a|≠0 and always >0

hence

For all ε >0, there exists δ >0 such that

3. **Continuity**: f(x) continuous at x=a

Case x=a: **|x-a|=0**

=> |f(a)-f(a)|= 0 <ε (automatically)

So by default we can remove (x=a) case.

Also from 1) it is understood: |**x-a|>0**

Hence suffice to write only:

f(x) is continuous at point x = a

For all ε >0, there exists δ >0 such that

For x->0, find limit L of

f(x)= (x³+5x)/x

1) guess L:

f(x)= x(x²+5)/x= x²+5

=> L= 5 when x->0

2) epsilon-delta Proof: find δ in function of ε such that:

|f(x)-5| < ε

|(x²+5)-5| <ε

|x|< √ε

Choose δ=√ε

For all ε, there is δ=√ε such that |x-0|< δ =>|f(x)-5|< ε

If ε=0.5, δ=√0.5=0.25

Rigorous Analysis epsilon-delta (ε-δ)

Cauchy gave **epsilon-delta** the rigor to Analysis, Weierstrass ‘* arithmatized*‘ it to become the standard language of modern analysis.

1) Limit was first defined by Cauchy in “** Analyse Algébrique**” (1821)

2) Cauchy repeatedly used ‘Limit’ in the book Chapter 3 “** Résumé des Leçons sur le Calcul infinitésimal**” (1823) for ‘derivative’ of f as the limit of

when i -> 0

3) He introduced ε-δ in Chapter 7 to prove ‘**Mean Value Theorem**‘: Denote by (ε , δ) 2 small numbers, such that 0< i ≤ δ , and for all x between (x+i) and x,

f ‘(x)- ε < < f'(x)+ ε

4) These ε-δ Cauchy’s proof method became the standard definition of **Limit of Function** in Analysis.

5) They are notorious for causing widespread discomfort among future math students. In fact, when it was first introduced by Cauchy in the **Ecole Polytechnique** Lecture, the French Napoleon top students booed at him and Cauchy received **warning** from the school.

Note 1: From the textbook ‘**Calculus**‘ (1980, USA):

“*If can’t understand the ‘ε-δ’ definition, just memorize it like a poem – at least better than saying it wrongly.*”

E.g. “**For all ε>0, there exists δ>0, …**”

Note 2: George Polya: “*The students are not trained in ‘ε-δ’, teaching them Calculus is like dropping these rules from the sky.*..”

Why Newton’s Calculus Not Rigorous?

…[1]

cancel x (≠0)from upper and below =>

…[2]

In [1]: we assume x ≠ 0, so cancel upper & lower x

But In [2]: assume x=0 to get L=5

[1] (x ≠ 0) contradicts with [2] (x = 0)

This is the weakness of Newtonian Calculus, made rigorous later by Cauchy’s ε-δ ‘Analysis’.