The following problem has appeared in 2013 January qualifying exam in Purdue University, which is publicly available here.
Problem 3. Let $\{a_k\}$ be sequence of positive numbers such that $a_n\to\infty$ as $n\to\infty$. Prove that the following limit exists $$ \lim_{k\to\infty}\int_{0}^{\infty} \frac{e^{-x}\cos(x)}{a_kx^2 + \frac{1}{a_k}} dx $$ and find it.
I have hardly come across to limits of sequences that involve definite integrals (in my undergraduate education so far), so this problem just seems insurmountable at the first glance. I would appreciate any hints.
One of the things that comes to mind is to use limit comparison test. For example, we can evaluate integrals such as $$\int_{0}^{\infty} e^{-x}\cos(x)=\frac{1}{2}$$ But for that we would have to bound the integrand somehow. One tempting thing is to interchange the integral and the limit, which would tell us that integrand is zero in the limit, but I highly doubt this is allowed here.
Looking forward to hear your thoughts.
P.S. I am not sure how to make the title informative for this post. Feel free to edit as you see fit.