if limit of f(x) as x approaches a exists, does the limit of |f(x)|exist as as x approaches a? vice versa?

Question

a) If lim f(x) exists, them lim |f(x)| exists

       x>a                         x>a

 

b) if lim |f(x)| exists, then lim f(x) exists 

       x>a                          x>a

 

true or false? 

i am not sure what the difference between these two statements is. 

or how to prove whether they are true or false. 

Answer 1

The answer to the first question is yes. This follows from the inverse triangle inequality which says that

||f(x)|-|A|| <= |f(x)-A|. So if we can make the difference f(x)-A very small we can also make the difference |f(x)|-|A| very small too and we actually have that limit of |f(x)| as x approaches a is equal to the absolute value of the quantity (limit of f(x) as x approaches a).

The answer to the second question is no. For example consider the function f(x)=1 for x>=0 and f(x)=-1 if x<0. Then |f(x)|=1 for all values of x which means that limit of |f(x)| as x goes to 0 is 1 but the limit of f(x) as x goes to 0 does not exist since from the left of 0 f(x)=-1 and from the right of 0 f(x)=1. This happens because f(x) has a jump discontinuity at x=0.