Question

Fill in the blanks to complete each of the following theorem statements:

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

Short answer

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

Step-by-step solution

Step 1. Given information. 

The given incomplete statement is the following.

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

Step 2. Explanation. 

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

Subtract L from all.

$$\begin{gathered} \left(L-\epsilon -L\right)<\left(f\left(x\right)-L\right)<\left(L+\epsilon -L\right) \\ -\epsilon <f\left(x\right)-L<+\epsilon \\ \left|f\left(x\right)-L\right|<\epsilon \end{gathered}$$

So the completed statement is the following.

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