I'll take a crack at this, but I'm a student so this is partly for my benefit too.
1) If we think of the sequence , what it would mean for the estimator to be asymptotically unbiased would be for the limit of this sequence to converge to . So, just going from what it means for a sequence of reals to converge, we would let be given and we have to find an N such that if n > N, then . This is easy, since that difference will be 0 for all n since the estimator is unbiased.
2) If the bias of the estimator is strictly positive, but tends to zero as the sample size tends to infinity, then this will be the case. The standard example would be which has a multiplicative bias of . Clearly as n goes to infinity this term goes to 1 which eliminates the bias.