Here is the question:
For the first integral I use the DCT, noting that converges to and that is bounded by 1. Then taking the limit inside, the integrand converges to 0 and therefore the upper integral is equal to 0. Is this correct? Any hints on the second one would be appreciated.
With DCT, the easiest here (this is a one-liner) is just to use the inequality , that follows from the binomial formula. And notice that the integrand converges to 0 as (polynomial vs. exponential).