Take $x_0=0$, then the integral
$$
\int_{-\infty}^\infty \sin\left(\frac{x}{\varepsilon}\right)dx
$$
diverges for every $\varepsilon>0$.  You can try a principal value for this.  There is a whole branch of analysis on "singular integral operators".  

Next try $x_0=1$, even the principal value diverges by oscillation:
$$
\int_{-t}^t \sin\left(\frac{x-1}{\varepsilon}\right)\;\frac{x}{x-1}\;dx =
\cos\left(\frac{1 + t}{\varepsilon}\right) \varepsilon + \text{Si} \left(\frac{1 + t}{\varepsilon}\right) - \cos\left(\frac{-1 + t}{\varepsilon}\right) \varepsilon + \text{Si} \left(\frac{-1 + t}{\varepsilon}\right)
$$

Here, $\varepsilon=1/10$:  

![$\varepsilon=1/10$][1]


  [1]: https://i.sstatic.net/ihjZ1m.gif