Past exam of the options II course of the University of Oxford / 2007 / D / D4 / ii / Solution

At each $x>0$, eventually $x<n$, and the stipulated power limit gives $f_n(x)\to e^{-x}\sin x$. At nonpositive $x$ the functions vanish. On their positive support,The supplied integrable exponential therefore dominates the sequence. Dominated convergence givesThe final improper integral is absolutely convergent, so agrees with its Lebesgue integral.