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Calculus: Differentiation and Integration, 1st Edition
book

Calculus: Differentiation and Integration, 1st Edition

by ICFAI
April 2024
Intermediate to advanced content levelIntermediate to advanced
710 pages
20h 50m
English
Pearson India
Content preview from Calculus: Differentiation and Integration, 1st Edition
Limits and Continuity 3-11
<
<
< <
1
4
1 1
2
3 3
4 9
3
2
3
3
2
4 9x
x
x
x d.
So we set d = min {1/4, 2e/3} to get the required result
x
x4 9
1
< e.
This completes the proof.
Example 3.9 Prove that
lim .
x
x
+
=
0
0
Solution: Let e > 0 be any real number.
Now, we need to find a d such that x <0 e whenever 0 < ½x½ < 0 + d
i.e., whenever
i.e., whenever
i.e., wh
x x
x x
x
< < <
< < <
<
ε δ
ε δ
ε
0
0
2
eenever 0 < <x δ.
Therefore, we set d = e
2
to get the required result.
This completes the proof.
3.1.2 Properties of Limits
Now that we have defined formally what a limit is and examined several examples. We state and prove
some properties that are useful for working with ...
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Publisher Resources

ISBN: 9789353948221