1) Does the absolute-value part have a derivative everywhere? (Think about where the graph of y = |x| has an "elbow"....)
2) For this part, just find the limits. Since |x| = -x for x < 0 and |x| = +x for x > 0, the limits will not have the same sign.
3) Can you have two one-sided limits in agreement if you have only one one-sided limit?