means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < δ, then |f(x) − L| < ϵ.
Thus we are able to use interchangeably the phrases
•
( Definition
Term) limx → a f(x) = L.
•
(Definition Description) For every ϵ > 0, there is some
δ > 0 such that, for all x, if 0 < |x −a| < δ,
then |f(x) − L| < ϵ.
We wish to decide which variations of a definition are "correct"
and give an equivalent definition.
Think
of a Equivalent Definitions as an If and Only If Theorem .
The phrase "Definition X is equivalent to
Definition Y" means you can use interchangeably the phrases
•
[Definition] Term (What is being defined)
•
[Definition] Description X (Details)
•
[Definition] Description Y (Details)
To show that two definitions X and Y for the same
Definition Term are equivalent we must show the following:
•
Satisfying Definition Description X
⇒ Satisfying Definition
Description Y.
•
Satisfying Definition Description Y
⇒ Satisfying Definition Description X.
Now if Definition X is not equivalent to
Definition Y for the same Definition Term , then at
least one of the following is false:
•
Satisfying Definition Description X
⇒ Satisfying Definition
Description Y.
•
Satisfying Definition Description Y
⇒ Satisfying Definition Description X.
Interpreting each of the above as a Theorem, the way to show
a Theorem is false is to construct a counterexample .
A counterexample is an object [construct, …] which
satisfies the hypotheses of the proposed Theorem, but does not
satisfy the conclusion[s] of the proposed Theorem.
Actual Definition of Limit Definition ACTUAL. (Actual, p. 96)
lim
x → a
f(x) = L
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < δ, then |f(x) − L| < ϵ.
Proposed Variations
For each of the proposed variations AA - OO of the actual
(Spivak) definition description of
lim
x → a
f(x) = L,
decide whether the proposed variation Definition XX is
equivalent to Definition Actual. Thus for each you must
think about the validity of the the two Theorems:
•
Satisfying Definition Description XX
⇒ Satisfying Definition
Description ACTUAL. If False, there is a counterexample.
•
Satisfying Definition Description ACTUAL
⇒ Satisfying Definition Description XX. If
False, there is a counterexample.
You may construct any counterexample graphically, by formula, or by a precise
description.
Definition AA.
lim
x → a
f(x) = L
means: For every ϵ > 0, there is some δ > 0 such that, for all x, 0 < |x −a| < δ, and |f(x) − L| < ϵ.
Definition BB.
lim
x → a
f(x) = L
means: For every ϵ > 0, there is some δ > 0 such that, for some x, 0 < |x −a| < δ, and |f(x) − L| < ϵ.
Definition CC.
lim
x → a
f(x) = L
means: For an ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < δ, then |f(x) − L| < ϵ.
Definition DD.
lim
x → a
f(x) = L
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < δ, |f(x) − L| < ϵ.
Definition EE.
lim
x → a
f(x) = L
means: For any ϵ > 0, there is a δ > 0 such that, for all x, if 0 < |x −a| < δ, then |f(x) − L| < ϵ.
Definition FF.
lim
x → a
f(x) = L
means: For any ϵ > 0, there is a δ > 0 such that, for all x, 0 < |x −a| < δimplies |f(x) − L| < ϵ.
Definition GG.
lim
x → a
f(x) = L
means: For any ϵ > 0, there is a δ > 0 such that, for all x, |f(x) − L| < ϵif 0 < |x −a| < δ.
Definition HH.
lim
x → a
f(x) = L
means: For any ϵ > 0, there is a δ > 0 such that, for all x, |f(x) − L| < ϵand 0 < |x −a| < δ.
Definition II.
lim
x → a
f(x) = L
means: For any ϵ > 0, there is a δ > 0 such that, for all x, |f(x) − L| < ϵwhenever 0 < |x −a| < δ.
Definition JJ.
lim
x → a
f(x) = L
means: For every ϵ > 0, there is a δ > 0 such that, for all x, |f(x) − L| < ϵfor 0 < |x −a| < δ.
Definition KK.
lim
x → a
f(x) = L
means: For an ϵ > 0, there is a δ > 0 such that, for all x, |f(x) − L| < ϵfor 0 < |x −a| < δ.
Definition LL.
lim
x → a
f(x) = L
means: For a δ > 0, there is an ϵ > 0 such that, for all x, |f(x) − L| < ϵprovided that 0 < |x −a| < δ.
Definition MM.
lim
x → a
f(x) = L
means: For all δ > 0, there is an ϵ > 0 such that, for all x, |f(x) − L| < ϵfor 0 < |x −a| < δ.
Definition NN.
lim
x → a
f(x) = L
means: For some δ > 0, there is an ϵ > 0 such that, for all x, if |f(x) − L| < ϵ, then 0 < |x −a| < δ.
Definition OO.
lim
x → a
f(x) = L
means: For some δ > 0, for all ϵ > 0, for all x, |f(x) − L| < ϵ if 0 < |x −a| < δ.
Footnotes:
1I borrow the
words Definition Term and Definition Description
from the html tags < DT > and < DD > .
File translated from
TEX
by
TTH,
version 4.04. On 22 Aug 2014, 03:51.