Limit of a Function
- [Play Video] 4½
minutes: Algebraic View of Limits. Example involving sums and
quotients.
- [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.
Suppose f(x) is a function of real numbers x and that
it is defined on an interval containing the number a.
[Limit of a Function]A function f(x) converges to a finite limit at the point
x = a if and only if 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 |
|
|
In the latter case, a limit L is said to exist and we
write
The in-line expression limx® a f(x) and the displayed expression
should both be read as the limit as x
goes to a of f(x). Here remember to read f(x) as f of x.
Continuity of a function f(x) at a number a
corresponds to the requirement that the
limit L = f(a). But it is possible for the limit L = limx® a f(x) to exist
and not equal f(a). See the chapter Slope
Approximation.
The rest of this chapter can be read lightly in the first instance.
The next sections are not needed in the immediately
following chapters.
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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
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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