MthT 430 Projects Chapter 5b Limits - Equivalent Definitions (sic)
Limits - Definitions (sic)
Definition. (Actual, p. 96)
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.
Some Correct and Incorrect Variations
Decide which, if any, of the Definitions A-N are equivalent to
the actual definition of
Definition A.
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.
Definition B.
means: For every ϵ > 0, there is some δ > 0 such that for some x, if 0 < |x −a| < δ, then |f(x) − L| < ϵ.
Definition C.
means: For every ϵ > 0, there is some δ > 0 such that for some x, 0 < |x −a| < δ, |f(x) − L| < ϵ.
Definition D.
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if |x −a| < δ, then |f(x) − L| < ϵ.
Definition E.
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if |f(x) − L| < ϵ, then 0 < |x −a| < δ.
Definition F.
means: For every ϵ > 0, there is some δ > 0 such that |f(x) − L| < ϵ, and 0 < |x −a| < δ.
Definition G.
means: For every δ > 0, there is some ϵ > 0 such that, for all x, if 0 < |x − a| < ϵ, then |f(x) − L| < δ.
Definition H.
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < ϵ, then |f(x) − L| < δ.
Definition I.
means: For every ϵ > 0, δ > 0, 0 < |x −a| < δ, |f(x) −L| < ϵ.
Definition J.
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < δ, then |f(x) − L| < 20 ϵ.
Definition K.
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < 20 δ, then |f(x) − L| < ϵ.
Definition L.
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < 10−6 δ, then |f(x) − L| < 106ϵ.
Definition M.
means: There is a number M such that for every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < 20 δ, then |f(x) − L| < M ϵ.
Definition N.
means: There is a number M such that for every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < δ/M, then |f(x) − L| < M ϵ.
Definition 106.
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < 106δ, then
|f(x) − L| < ϵ.
106 ⇒ Actual:
Fix ϵ > 0. By 106, there is an δ > 0 such
that, for all x, if 0 < |x −a| < 106δ, then
|f(x) − L| < ϵ.
If 0 < |x −a| < δ, then 0 < |x −a| < 106δ,
so |f(x) − L| < ϵ.
Actual ⇒ 106:
Fix ϵ > 0. By Actual, there is an η > 0 such
that, for all x, if 0 < |x −a| < η, then
|f(x) − L| < ϵ. Let δ = 10−6η. Then δ > 0 iff η > 0. For all x, if
0 < |x − a| < 106δ = η, |f(x) − L| < ϵ.
From actual student papers:
Definition St1.
means: For every ϵ > δ, the limit of f(x) as x goes to a is L.
Definition St2.
means: There exists a δ > 0 and ϵ > 0 such that |x − a| < δand |f(x) −L| < ϵ.
Definition St3.
means:
Therefore f(a) = L.
Definition St4.
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.
Definition St5.
means: There exists some ϵ > 0 such that |x − a| = δδ > 0 as |ϵ−δ| gets "small" f(x) = L.
Definition St6.
means: There exists a f(ϵ) ≤ f(a) ≤ f(δ) for some ϵ,δin the domain of f(x). I don't remember!!
Definition S6.
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.
Definition St7.
means: For every ϵ > 0, there exist δ > 0 such that if 0 < |x −a| < δ, then |f(x) − f(a)| < ϵ.
Definition St8.
means: For every ϵ > 0, there is some δ > 0 such that, for all x, if 0 < |x −a| < 0, then |f(x) − L| < ϵ.
Definition St9.
means: Given an ϵ > 0, but otherwise as small as we like, we can find that δ > 0 such that 0 < |x −a| < δ, given |f(x) − L| < ϵ.
Definition St10.
means: The limit of a function as it approaches a.
Definition St11.
means: If f(x) is within ϵof L is within δof a. So the closer we desire our f(x) to be to L the smaller we must choose an δ, until it is clear that f(x) approaches L as x approaches a.
Definition St12.
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.
Definition St13.
means: A continuous function has a limit at a if |f(x) − f(a)| < ϵ, ϵ > 0 when |x− a| < δfor some δ > 0 and the limit L = f(a).
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