I want to evaluate $$\lim_{k \to \infty} \int_{(0,1)}\frac{(1-x)^k \cos(k/x)}{\sqrt{x}}dx$$
where the integral is Lebesgue-integral. Attempt:
Note first that for $x \in (0,1)$, we have $\lim_k \frac{(1-x)^k \cos(k/x)}{\sqrt{x}} = 0$. Also,
$$\left|\frac{(1-x)^k \cos(k/x)}{\sqrt{x}}\right| \leq 1/\sqrt{x}$$
and by monotone convergence theorem $$\int_{(0,1)}1/\sqrt{x}dx = \lim_n \int_{(1/n,1)} x^{-1/2}dx = \lim_n (2-2\sqrt{1/n})=2$$
so the dominated convergence theorem allows us to interchange integral and limit and we conclude that
$$\lim_{k \to \infty} \int_{(0,1)}\frac{(1-x)^k \cos(k/x)}{\sqrt{x}}dx = \int_{(0,1)}0 dx = 0$$
Is this correct?
To simplify it a bit, note that the absolute value of the $k$th integral is bounded above by
$$\int_{(0,1)}\left |\frac{(1-x)^k \cos(k/x)}{\sqrt{x}}\right |\,dx \le \int_{(0,1)}\frac{(1-x)^k }{\sqrt{x}}\,dx$$
On the right the integrands are $\le 1/\sqrt x \in L^1.$ Furthermore these integrands $\to 0$ pointwise. The DCT then tells us the limit is $0.$