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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
Lets see what wo ul d c han ge in the calculat io n in Example 4.5.11 if, while using the same function,
we consider the limit at a different point. Lets repeat the procedure as closely as possible.
EXAMPLE 4.5.12
Prove that lim
x1
ðx
2
+ 3x + 1Þ= 5:
Proof: The goal is to prove that for every given positive number ε there exists a positive number δ such that if x
satisfies the inequality |x 1| < δ,then|(x
2
+3x +1) 5| < ε. Thus, we need to determine how the requirement
|(x
2
+3x +1) 5| < ε will influence |x 1|. Lets start by performing some algebra:
x
2
+ 3x + 1Þ 5 j= jx
2
+ 3x 4 j= x 1Þðx + 4Þj= jx 1 jjx + 4 j:
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Publisher Resources

ISBN: 9780123822178