Nevermind, figured it out.
Great. For the benefit of others who may wonder about this, note that, if x is close to 0, 3x-1 will be negative and 3x+1 will be positive. That is, |3x-1|= -(3x-1)= -3x+ 1 and |3x+1|= 3x+1. |3x-1|- |3x+1|= -3x+1- 3x-1= -6x. -6x/x= -6.
You are correct that just taking x= 0 gives the indeterminate form 0/0 but that does not contradict the fact that that the limit is -6.