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Q. 4

Expert-verified

Calculus

Found in: Page 777

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Fill in the blanks to complete each of the following theorem statements:
For $ε > 0, f (x) \in (L - ε, L + ε)$ if and only if $< ε .$

For $ε > 0, f (x) \in (L - ε, L + ε)$ if and only if role="math" localid="1648287054505" $|f (x) - L| < ε .$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

The given incomplete statement is the following.

For $ε > 0, f (x) \in (L - ε, L + ε)$ if and only if $< ε .$

Step 2. Explanation.

$f (x) \in (L - ε, L + ε)$ can be elaborate as $L - ε < f (x) < L + ε$

Subtract L from all.

$(L - ε - L) < (f (x) - L) < (L + ε - L) - ε < f (x) - L < + ε |f (x) - L| < ε$

So the completed statement is the following.

For $ε > 0, f (x) \in (L - ε, L + ε)$ if and only if $|f (x) - L| < ε .$

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