Series expansion of function with 0/0 behaviour near origin

Question

I am interested in the behaviour of $$f(x)=\frac{1}{x}exp \left [-a \left ( \frac{b-x}{x} \right)^2 \right ]$$ near $x=0$. Evidently $f(x) \rightarrow 0$ as $x \rightarrow 0$, because the $x$-denominator in the argument of the exponential will cause $f$ to decrease very rapidly with decreasing $|x|$. An expansion, or even limit calculation, would be welcomed.

Answer 1

We have that assuming $a>0$

$$e^{\left [-a \left ( \frac{b-x}{x} \right)^2 \right ]}=e^{-\frac{ab^2}{x^2}}e^{\frac{2ab}{x}}e^{-a}\sim e^{-\frac{ab^2}{x^2}}$$

therefore

$$f(x)\sim \frac1x e^{-\frac{ab^2}{x^2}} \to 0$$

indeed wlog $x>0$ by $y=\frac1x$

$$\frac1x e^{-\frac{ab^2}{x^2}}=\frac{y}{e^{ab^2y^2}}\to 0$$