Approximation for $\int dx\, f(x) / (x-x_0) \exp (-g(x)^2 / (x-x_0)^2)$

Question

Is there an approximation that can be made for

$$\int dx\, \frac{f(x)}{x-x_0} \exp \left(- \frac{g(x)^2}{(x-x_0)^2} \right)$$,

where $g(x_0) = 0$? I know that a thin Gaussian can be approximated as a Dirac delta function, but this seems to be slightly different. Is there a systematic expansion that can be made?