Codification of the Limit Concept

We now take a leap and formalized or codify the decimal pattern in the above examples. 

Definition of Limit of a Function 
and Notation for it

Suppose f(x) is a function of real numbers x and that it is defined on an interval  containing the number a. 

A function f(x) converges to a finite limit L at the point x = a if and only if (when and only when)  there is a real number L such that for every integer n, there is an m such that

|x-a| < d = 1 2 1 10m     implies    |f(x)-L| < e = 1 2 1 10n

Here  d (delta) and e (epsilon) are Greek Letters. Use or think of letters D and E in place of them if you like.

In the latter case, a limit L is said to exist and we write

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Infinite Limits.

We write  lim  _{x® c} g(x) = oo and say the value of the limits is oo when and only when for every whole number M > 0,  there is a whole number k such that 

g(x) > M whenever  |x - c| < 10^-k

And we write  lim  _{x® c} g(x) = -oo and say the value of the limits is oo when and only when for every whole number M > 0,  there is a whole number k such that 

g(x) < - M whenever  |x - c| < 10^-k

 When we say or write that a limit   lim  _{x® c} g(x)  has an infinite value (or approaches plus or minus infinity),  we are describing   the behavior of g(x) as  x® c  but we are not giving a finite number L as the limit. Thus a finite limit does not exist.  

Remark (Technical Trap):  

Now in speaking of limits, mathematics follows the technical convention that a limit 

 lim  _{x® c} g(x) 

exists when and only when there is a real number L such that 

L =  lim  _{x® c} g(x) .   

Odd conventions like this in calculus provides  a test of precision reading skills in calculus.

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0. Triangle Inequality
0 Inequalities
0. Solving Inequalities
1. Distance+Midpt Formulas
1. Function Domains
1.Polynomial Domain+Range
1. Fn:  Linear Combinations
1. Fn Composition II
1. Fn Composition I
1. Solving y**n = x**m
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
3. Derivative Motivation
3.  Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x)  (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3.  Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
4.  Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5  Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere

To Learn More, visit Volumes 2 and 3.

Advanced Topics

Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity

To Learn More: Visit Real Analysis.

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