"Things don't cancel like they should" because the numerator is NOT 0 when x= 1. What does that tell you about the limit?
As for the second, |x| "changes formula" at x= 0. |3x- 2| changes formula when 3x- 2= 0 and |3x+2| changes formula when 3x+ 2= 0. What are the solutions to that? Those values of x divide the real number line into three intervals.