How to show that $\lim_{\varepsilon \rightarrow 0^+} \int_0^\infty \frac{\varepsilon}{\varepsilon^2+x}\sin(1/x) \, dx=0$? So I tried integration by parts and substitution but I couldn't bound the integral on the inside so that the limit would tend towards zero. Any hints on how to do this?
Here is the work I have done so far
\begin{align}
\int_0^t \frac{\varepsilon}{\varepsilon^2+x} \sin(1/x) \, dx & = \int_0^t \frac{\varepsilon x}{\varepsilon^2+x^2}\sin(1/x^2) \, dx \\
&= \left.\arctan \left(\frac x {\varepsilon}\right) x\sin(1/x^2)\right|_0^t-\int_0^t\arctan \left(\frac x \varepsilon \right) \frac d {dx}(x\sin(1/x^2)) \, dx\\
&=-\int_0^t\arctan\left(\frac x \varepsilon \right)\frac{d}{dx}(x\sin(1/x^2)) \, dx
\end{align}
 A: Using the change of variable $u= \frac{\varepsilon}{x}$ i.e $dx = -\frac{\varepsilon}{u^2}du$ leads to
$$ \int_0^\infty \frac{\varepsilon}{\varepsilon^2+x}\sin(1/x) \, dx=\int_0^\infty \frac{\sin(x/\varepsilon)}{x^2+x/\varepsilon}\, dx $$
Use the convergence dominated theorem as follow

Fix $0<\varepsilon \le 1$

Let $$f_\varepsilon(x) =  \frac{\sin(x/\varepsilon)}{\varepsilon^2+x/\varepsilon}$$
then, for every $x>0$
$$\lim_{\varepsilon \rightarrow 0}f_\varepsilon(x) =  \frac{\sin(x/\varepsilon)}{x^2+x/\varepsilon} = 0 $$
Since $|\sin(x/\varepsilon)|\le 1$

*

*For $x>1$ we have that

$$ |f_\varepsilon(x)| = | \frac{\sin(x/\varepsilon)}{x^2+x/\varepsilon}| \le \frac{1}{x^2}$$
Since $ |\sin a|\le 1 $.


*For $0<x\le 1$  using the fact that $|\sin a|\le |a|$ we have

$$ |f_\varepsilon(x)| = |\frac{\sin(x/\varepsilon)}{x^2+x/\varepsilon}| \le\frac{x/\varepsilon}{x^2+x/\varepsilon} \le 1$$


*

*+2) imply that
$$|f_\varepsilon(x)|\le \frac{1}{x^2}\chi_{(1,\infty)}(x)+\chi_{(0,1)}(x) $$

Setting $$\phi(x)=\frac{1}{x^2}\chi_{(1,\infty)}(x)+\chi_{(0,1)}(x) $$
Where $\chi_B$ is the characteristic function of the set B i.e
$\chi_B(x) = 1$ if $x\in B$ and $\chi_B(x) = 0$ elsewhere.
Then, it is easy to check that, $\phi \in L^1(0,\infty)$.

Conclusion $f_\varepsilon(x)\to 0$ as $\varepsilon \to0$ and $|f_\varepsilon(x)|\le \phi(x)$ with $\phi \in L^1(0,\infty)$.
hence by convergence dominated theorem,

$$\lim_{\varepsilon \to 0} \int_{0}^{\infty} f_\varepsilon(x) dx =0$$
A: Here is another and more direct way to come across this issue. Still we use the convergence dominated theorem  as follows
Set $ h(x,\varepsilon) = \frac{\varepsilon}{\varepsilon^2+x}\sin(1/x) \, $
Obviously for every $x>0$
$$\lim_{\varepsilon \rightarrow 0^+}  \frac{\varepsilon}{\varepsilon^2+x}\sin(1/x) =0$$
Let set
 $$g(x) = \sup_{\varepsilon >0} \{|h(x,\varepsilon)|\} = \sup_{\varepsilon >0} \{\frac{\varepsilon}{\varepsilon^2+x}|\sin(1/x)|\} = \frac{|\sin(1/x)|}{2\sqrt{x}} $$
with $g\in L^1(0,\infty)$
Thus for very $\varepsilon>0$,
$$|h(x,\varepsilon)|\le g(x) =\frac{|\sin(1/x)|}{2\sqrt{x}}    $$
with $g\in L^1(0,\infty)$ hence by convergence dominated theorem, 
$$\lim_{\varepsilon \rightarrow 0^+} \int_0^\infty \frac{\varepsilon}{\varepsilon^2+x}\sin(1/x) \, dx=0$$

Indeed,Let prove that $g(x)= \frac{|\sin(1/x)|}{2\sqrt{x}}$ and $g\in L^1(0,\infty)$ 

$$ \frac{\partial }{\partial \varepsilon}\left(\frac{\varepsilon}{\varepsilon^2+x}\right) =  \frac{x-\varepsilon^2}{(\varepsilon^2+x)^2} = 0\Longleftrightarrow \varepsilon = \pm\sqrt{x}$$
Therefore for $ \varepsilon\in (0,\infty)$ the only critical value of the function $\varepsilon \mapsto \frac{\varepsilon}{\varepsilon^2+x}$ is given by, $\varepsilon = \sqrt{x}$. Moreover, 
we have that 
$$ \lim_{\varepsilon \rightarrow {\infty,}{0^+} } \frac{\varepsilon}{\varepsilon^2+x} \, =0$$
we then, conclude that, $ \varepsilon = \sqrt{x}$ is the global maximum of the function $\varepsilon \mapsto \frac{\varepsilon}{\varepsilon^2+x}$ i.e 
$$ \sup_{\varepsilon>0}\{\frac{\varepsilon}{\varepsilon^2+x}\} =\frac{\sqrt{x}}{\sqrt{x}^2+x}=\ = \frac{1}{2\sqrt{x}}\, $$
Now let show that $g\in L^1(0,\infty)$.
$$ \int_{0}^{\infty}g(x) dx =\int_{0}^{\infty}\frac{|\sin(1/x)|}{2\sqrt{x}} dx =\int_{1}^{\infty}\frac{|\sin(1/x)|}{2\sqrt{x}} dx +\int_{0}^{1}\frac{|\sin(1/x)|}{2\sqrt{x}} dx\le \int_{1}^{\infty}\frac{1}{2x^{\frac{3}{2}}} dx +\int_{0}^{1}\frac{1}{2\sqrt{x}} dx <\infty. $$
Where in the first integral we used $|\sin a|\le |a|$ , in the second we used $|\sin a|\le 1$
