Appetizers and Lessons for Mathematics and Reason 
Thank you for visiting  www.whyslopes.com :1200+ pages.  Site coverage of complex numbers is unique

2. Decimal View of Limits
Back ] Section Entrance ] Up ] Next ]
Starter & Warm Up Lessons ] 1. Usual Review/Starter Lessons ] 2. Limits [13] ] 3. Differentiation Rules[28] ] 4. Applications of Derivatives [5] ] 5. Definite Integrals - Preview [5] ] 6. Integration Applications [6] ] Advanced Material ]

More Calculus

Vol 2, Three Skills for Algebra covers many  topics in algebra and logic that students starting calculus should have mastered or will have to master sooner or later. Also includes arithmetic review problems to catch common mistakes.  
 Vol. 3, Why Slopes & More Maths, gives starter lessons for differential and integral calculus.

Starter Guide (Views)
Real Player Videos

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

Starter & Warm Up Lessons
1. Usual Review/Starter Lessons
2. Limits [13]
3. Differentiation Rules[28]
4. Applications of Derivatives [5]
5. Definite Integrals - Preview [5]
6. Integration Applications [6]
Advanced Material

 


Decimal Insights on Limits, Continuity, Convergence

Decimal and decimal-free error-control perspectives of continuity, limits and Cauchy sequences are given below. These perspectives is followed by comments on math education.

Continuity and Unlimited Error Control

Limits and continuity in calculus may be described geometrically, that is, intuitively and informally, or more precisely in terms of say epsilons and deltas. The roles of epsilon and delta below are played by E > 0 and D > 0.

Imagine for instance we want to compute a function f(x) at the point x = A accurately. So we can ask the error control question how close must x be to A in order for f(x) to agree with f(A) to say k-decimal places. The answer might be that x must agree with A to m decimal places. In some computational problems, this answer for a specified number k of decimal places may be all that is needed. But in other situations, we want in practice or in principle, unlimited error control. Here we may want to say for any k, there is an m such f(x) will agree with f(A) to k decimals if x agrees with A to m decimals. Unlimited error control offers motivation and a perspective on the discussion of continuity.

Now will say that f(x) is continuous at x = A if for each whole number k, there is a number m such the limit f(a) and the value of f(x) will agree to k-decimals whenever the number x agrees with the value of a to m decimal places. Continuity here represents the concept of unlimited error control in decimal computations.

More generally, we can ask (following Cauchy), given an error control target E > 0, how close must x be to A for the difference of f(x) and f(A) to be less than E in magnitude? The answer follows by obtaining a number D with the property that if |x-A| < d then |f(x) - f(A)| < E.

Without reference to decimals we can say that f(x) is continuous at x = A if for every error control tolerance E > 0, there is a number D > 0 such that whenever |x-A| < d then |f(x)- f(A)| < E. Here continuity at x="A" corresponds to the idea of unlimited error control at x="A." This second concept is decimal free. It is traditional to use epsilons and deltas in place of E> 0 and D > 0.

Limits

We will say that a number L is the limit of a function f(x) as x approaches A if one of the following conditions hold:

  • (Decimal Perspective): For every whole number k, there exist a whole number m such f(x) will agree will L to k decimal places if x agrees with A to m decimal places.
  • (Decimal Free Perspective): For every positive number E > 0, there exists a positive number D> 0 such that if |x-A| < d then |f(x)- f(A)| < E.

Both conditions are equivalent. Each implies the other. Why or how depends on how you think of (or represent) the real numbers. For most people, assuming that real numbers are represented by signed decimal expansions (infinite or finite) is sufficient. Modern mathematics has alternate decimal (or base) -free representations of real numbers.


 

 

Calculus Students:  Hire the site author, as an online tutor.  Invitations to group lessons on popular or much needed topics may follow.   Site Reviews may serve as references.  Online whiteboards with  voice and real-time writing make online tutoring easy and efficient - board content printable.  Text or written work scanned or saved to a  pdf file may be  uploaded  for discussion in the whiteboard.  The first lesson is free to show what is offered. Bon Appetite

www.whyslopes.com

Parents: Help your Child/Teen Learn

Online Volumes
 
(orders)
1,  Elements of Reason. 1996
1A. Pattern Based Reason  1995
1B. Math Curriculum Notes 1996
2. Three Skills for Algebra  1995
3 .Why.Slopes.&
.More.Math.1995

Math How-TOs etc  2008
1. Arithmetic
2. Algebra 
3. More Algebra 
4. Geometry  
5. More Geometry
6. Calculus

Site Description/Reviews  by 3rd parties

Site  Math Lessons
1. Arithmetic Flash Videos  11-2008
2.  Algebra Videos (to appear)
3. Fractions and More 
4.. 
Solving Linear Equations  04-2005
5. Euclidean-Geometry To Complex No.s 
6.  Analytic Geometry/Functions 2006
7.  Number Theory. 2006-7
8.
  Exponents, Radicals & logs. 2008
9 Calculus  2005
10..Real  Analysis 1995
11 Electric Circuits Etc  2007
12. .Algebra, Odds & Ends, HS level-2001
13.Maps, Plans,  Similarity &Trig, with
Complex   Numbers
, 12-2009. 

For Math Instructors/Tutors/
Curriculum Committees


1. K0-11Applied Math Program Outline  
2. Mathematics education  essays 
3. LAMP - an earlier applied math program.
4.
(150 pages)

www.whyslopes.com/search

Visitors:  Ask a question by email if you cannot find what you need in www.whyslopes.com  for high school or college maths courses - answers will be added to site content.

 Back ] Up ] Next ] [Top of this Page]  
Mathematics Education Consulting and Private (Online) Instruction available

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by
email,  errors in mathematics or grammar or terminology to site author
If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,  
Mathematics Consultant/Tutor/Instructor, All Rights Reserved.