Chapter 1.3, Problem 1C

(a)

To determine

Whether given statement is true or false.

‘If $x\ne c$ then $|x-c|$ is strictly greater than zero’

(b)

To determine

Whether given statement is true or false.

‘If $|x-c|$ is strictly greater than zero then $x\ne c$ ’

(c)

To determine

Whether given statement is true or false.

‘x is a solution of $0<|x-c|<\delta$ if and only if $c-\delta <x<c+\delta$ ’

(d)

To determine

Whether given statement is true or false.

‘If $0<|x-c|<\delta$ then $x\in (c-\delta ,c)\cup (c,c+\delta )$’

(e)

To determine

Whether given statement is true or false.

‘If $|f(x)-l|<\in$ then $l-\in <f(x)<l+\in$ ’

(f)

To determine

Whether given statement is true or false.

‘If $f(x)\in (L-\in ,L+\in )$ , then $0<f(x)<|L+\in |$’

(g)

To determine

Whether given statement is true or false.

‘The fact that $0<|x-3|<0.25$ guarantees that $|(2x-1)-5|<0.5$ proves that $\underset{x\to 3}{\lim }(2x-1)=5$ .’

(h)

To determine

Whether given statement is true or false.

‘ $\underset{x\to 3}{\lim }(2x-1)=5$ means that for all $\delta >0$ there is some $\in >0$ such that if $0<|x-c|<\delta$ then $|(2x-1)-5|<\in$ ’