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  14 Limits & Error Control (V)  Back ] Home ] Next ]    

Chapter 14
Limits, Error Control and Continuity

  • [Play Video]  4½ minutes: Algebraic View of Limits. Example involving sums and quotients.
  • [Play Video]  2½   minutes: Algebraic Properties of Limits I.
  • [Play Video] 2¼ minutes: Algebraic Properties of Limits II.
  • [Play Video]  5½ minutes: Limits and Error Control for Linear Expressions
  • [Play Video]  2¾ minutes: Error Control to N decimal Places, say 5 or 10. 
  • [Play Video]  3¼ minutes:  Limits as Error Control for an unlimited number of decimal places. 

Error control for the evaluation of functions y = f(x) provides a simple context and motivation for continuity and convergence.

Continuity at Point

To explain the idea of continuity of a function y = f(x) at a point x = a, we ask the following error-control question with b = f(a): to what number m of places should the decimal expansions of x and a agree, for the decimal expansion of the number f(x) to agree with that of b = f(a) to n-decimal places? That is, given a whole number n, is there an m such that
|x-a| < d = 1
2
· 1
10m
    implies    |f(x)-f(a)| < e = 1
2
· 1
10n
    (?)
An affirmative answer requires that agreement of x with a to m decimal places implies the agreement of f(x) with f(a) to n decimal places. An affirmative answer says unlimited accuracy and error control is possible at x = a.

The Greek letters d (delta) and e (epsilon) above are employed here in accordance with tradition of some (not all) calculus texts. For simplicity, the error control tolerances e and d in the first instance here and below, may be restricted to be numbers of the form [1/2] ·10-k = [1/2] [1/(10k)]. The decimal free discussion of error control and continuity dispenses with this requirement.

We say a function f(x) is continuous at a point x = a if and only if unlimited error control is possible there. More formally, we state the following definition.

Theorem 14.1 [Continuity at a Point] If f(x) is a real-valued function of a real number x in an interval [c,d], and a is a number in the interval [c,d] then the function f is said to be continuous at the number x = a if and only if the following holds. If for every n, there exist an m such that
|x-a| < d = 1
2
· 1
10m
    implies    |f(x)-f(a)| < e = 1
2
· 1
10n
·

Decimal-Free Form

The decimal-free description or definition of continuity at a point x = a is as follows.

[Continuity at Point] If f(x) is a real-valued function of a real number x in an interval [c,d], and a is a point in the interval [c,d] then the function f is said to be continuous at x = a if and only if the following holds: For every e1 > 0, there exist a d1 > 0 such that
|x-a| < d1     implies    |f(x)-f(a)| < e1
It is easily shown that the decimal-free and decimal-based definitions are equivalent. The proof of equivalence, better left to a second reading of this work, follows.

Proof of Equivalence.

To show the decimal-based description implies the decimal-free description of continuity, observe the following. First given e1 > 0, there is an n > 0 such that e1 > [1/2]·[1/(10n)] = e. The decimal-based requirement for continuity now is satisfied for some d = [1/2] ·[1/(10m)]. So the decimal-free version holds with d1 = d = [1/2]·[1/(10m)].

Conversely, the other way that is, to show the latter decimal-free form implies the decimal-based description of continuity, observe the following. Given m > 0, let e1 = e = [1/2] ·[1/(10m)]. Then choose d1 > 0 so that the decimal-free requirement is satisfied. The decimal-based version is then satisfied if m > 0 is selected so that d1 ³ d = [1/2] ·[1/(10m)].

 

Why Slopes
and
More Math

understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7

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Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review  (optional)
9 On Calculus Studies
11 Slope of Slope
13  Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General  (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17  What is Area
18 Integration
18 Area Calculation
18  Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc


If  you like Volume 3  you may also like  Three Skills for Algebra , Exponents & Radicals Exactly,  complex numbers, Euclidean Geometry , More Calculus and  Pattern Based Reason  as well. 

Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity  Motions
10 Slopes without Units.
10 Units & Slopes
10  Units in Cost vs. Quantity
10  How Units  Appear
10 Unit  Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units

Enriched material: The Appendices of Volume 3 are located in the Real  Analysis  Area.

Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side Theorem
Integration & Lipschitz
 Continuity


These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
 Range Theorem
is a postscript,
not in printed version.


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