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The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
Since j x 1j< δ, we have j
x
3
1
x
2
1
3
2
j<
j2x + 1j
2jx + 1j
δ: We need to estimate the fraction
j2x + 1j
2jx + 1j
: The choice of
δ = minimum 1,
2
5
ε

implies that δ 1andδ
2
5
ε: Thus jx 1j< δ 1: So, 1 < x 1 < 1, that is,
0 < x < 2: To get an estimate for the numerator, we notice that 0 < 2 x < 4and1< 2x + 1 < 5: As an
estimate for the numerator, we have 1< x + 1 < 3: Therefore,
j2x + 1j
2jx + 1j
<
5
2 × 1
=
5
2
: This allows us to
conclude that
x
3
1
x
2
1
3
2
<
j2x + 1j
2jx + 1j
δ <
5
2
δ
5
2
×
2
5
ε = ε:
This inequality implies that the choice of δ = minimum 1,
2
5
ε

is indeed correct.
20. Let δ = 0:9: Then jx 2j< δ = 0:9: This implies 0:9 < x 2 < 0:9,
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Publisher Resources

ISBN: 9780123822178