For your first question, the key trick is to write . You know that , suitably normalized, converges in distribution. As for , the LLN shows that converges almost-surely (and hence in distribution) to 2.

For the second one, do you have an intuition of what looks like? Once you have it, prove that converges in distribution to you-guessed-what, using the cumulative distribution function (which I prefer) or the MGF. Then deduce the result about .

By the way, the MGF is . You can simplify this expression (sum of terms in a geometric sequence...).