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Expert-verified Found in: Page 777 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Fill in the blanks to complete each of the following theorem statements: For $\epsilon >0,f\left(x\right)\in \left(L-\epsilon ,L+\epsilon \right)$if and only if $\overline{)}<\epsilon .$

For $\epsilon >0,f\left(x\right)\in \left(L-\epsilon ,L+\epsilon \right)$if and only if role="math" localid="1648287054505" $\overline{)\left|f\left(x\right)-L\right|}<\epsilon .$

See the step by step solution

## Step 1. Given information.

The given incomplete statement is the following.

For $\epsilon >0,f\left(x\right)\in \left(L-\epsilon ,L+\epsilon \right)$if and only if $\overline{)}<\epsilon .$

## Step 2. Explanation.

$f\left(x\right)\in \left(L-\epsilon ,L+\epsilon \right)$can be elaborate as $L-\epsilon

Subtract L from all.

$\left(L-\epsilon -L\right)<\left(f\left(x\right)-L\right)<\left(L+\epsilon -L\right)\phantom{\rule{0ex}{0ex}}-\epsilon

So the completed statement is the following.

For $\epsilon >0,f\left(x\right)\in \left(L-\epsilon ,L+\epsilon \right)$if and only if $\left|f\left(x\right)-L\right|<\epsilon .$ ### Want to see more solutions like these? 