<IMG SRC="http://ilorentz.org/beenakker/MO/BesselK.png" WIDTH="400" />

The blue curve is the desired integral $\int_ 0^{\infty}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$. The exact small-$a$ asymptotics is $ae^{-1/a}$.