Evaluate $\lim_{n\to\infty}\int_0^k \frac{1-\sin(\frac{n}{x})}{\sqrt{x^2+1/n}}dx\;$ I'm having troubles evaluating the following limits:
$$\lim_{n\to\infty}\int_0^1 \frac{1-\sin(\frac{n}{x})}{\sqrt{x^2+1/n}}dx\;.$$
$$\lim_{n\to\infty}\int_1^{2011} \frac{1-\sin(\frac{n}{x})}{\sqrt{x^2+1/n}}dx\;.$$
I tried,for instance, to use the dominated convergence theorem, for instance noticing that
$$|\frac{1-\sin(\frac{n}{x})}{\sqrt{x^2+1/n}}|\le \frac{2}{\sqrt{x^2+1/n}}$$
but then in 0 the integral diverges...
Thank you in advance!
 A: $$
\begin{align}
\lim_{n\to\infty}\int_0^1\frac{1-\sin(\frac{n}{x})}{\sqrt{x^2+1/n}}\mathrm{d}x
&=\lim_{n\to\infty}\int_1^\infty\frac{1-\sin(nx)}{\sqrt{1+x^2/n}}\frac1x\,\mathrm{d}x\tag{1}\\
&=\lim_{n\to\infty}\int_{1/\sqrt{n}}^\infty\frac{1-\sin(n^{3/2}x)}{\sqrt{1+x^2}}\frac1x\,\mathrm{d}x\tag{2}\\
&=\lim_{n\to\infty}\int_{1/\sqrt{n}-\delta}^\infty\frac{1+\sin(n^{3/2}x)}{\sqrt{1+(x+\delta)^2}}\frac1{x+\delta}\,\mathrm{d}x\tag{3}\\
&=\lim_{n\to\infty}\int_{1/\sqrt{n}}^\infty\frac{1+\sin(n^{3/2}x)}{\sqrt{1+x^2}}\frac1x\,\mathrm{d}x\\
&+O\left(\frac1n\right)\tag{4}\\
&=\lim_{n\to\infty}\int_{1/\sqrt{n}}^\infty\frac1{\sqrt{1+x^2}}\frac1x\,\mathrm{d}x\\
&+O\left(\frac1n\right)\tag{5}\\
\end{align}
$$
where $\delta=\pi n^{-3/2}$ and $(5)$ is the average of $(2)$ and $(4)$. The $O\left(\frac1n\right)$ in $(4)$ comes from the size of $\delta$ and the derivative of $\frac1{\sqrt{1+x^2}}\frac1x$. The integral in $(5)$ grows like $\frac12\log(n)$. Thus,
$$
\lim_{n\to\infty}\int_0^1\frac{1-\sin(\frac{n}{x})}{\sqrt{x^2+1/n}}\mathrm{d}x=\infty\tag{6}
$$
