True or False: limits

Question

a) if limit of f(x) as x approaches a exists, then limit of  |f(x)| as x approaches a exists. 

is this true or false? why or why not?

 

b) if limit of |f(x)| as x approaches a exists, then limit of f(x) as x approaches a exists 

true or false?

 

I'm not sure what the difference between these two statements is. 

Also, how do i prove either is true or false? 

Answer 1

The b statement is just the converse of the a statement, if you remember geometry.

 

They are both TRUE.

 

If limit of f(x) as x approaches a exists, then

lim_x→af(x)    = L, where L is a finite number

lim_x→a|f(x)| = |lim_x→af(x)|    {Function of a limit = limit of the function; |.| is a function}

                    =|L|, a non negative finite number

 

So the statement "if limit of f(x) as x approaches a exists, then limit of |f(x)| as x approaches a exists" is TRUE.

 

Conversely, if limit of |f(x)| as x approaches a exists, then

 

lim_x→a|f(x)| = L where  L is a non negative finite number

|lim_x→af(x)|= L

lim_x→af(x)   = L or −L, a finite number.

 

So the statement "if limit of |f(x)| as x approaches a exists, then limit of f(x) as x approaches a exists" is TRUE.