Q. 3 - Calculus

Question

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

For $\delta >0, x\in (c-\delta ,c)\cup (c,c+\delta )$if and only if $0<\underline{ }<\delta .$

Step-by-Step Solution

Answer

 For $\delta >0, x\in (c-\delta ,c)\cup (c,c+\delta )$if and only if $0<\underline{|x-c|}<\delta .$

Step 1. Given information.

The given incomplete statement is the following. 

For $\delta >0, x\in (c-\delta ,c)\cup (c,c+\delta )$if and only if $0<\underline{ }<\delta .$

Step 2. Explanation.

$x\in (c-\delta ,c)\cup (c,c+\delta )$can be elaborate as $c-\delta <x<c+\delta .$

Subtract c from all.

$$\begin{gathered} \left(c-\delta -c\right)<\left(x-c\right)<\left(c+\delta -c\right) \\ -\delta <x-c<+\delta \\ \left|x-c\right|<\delta \end{gathered}$$

So the completer statement is the following.

For $\delta >0, x\in (c-\delta ,c)\cup (c,c+\delta )$if and only if $0<|x-c|<\delta .$