Suppose I have a Laplace transform,

$$ F(s) = \int_0^{\infty}dx\ f(x)e^{-s x} \ . $$

I know that

$$ F(s) \approx e^{A/(4s)} $$

(for $s$ real) where $A$ is very large, and I want to estimate $f(x)$. I suspect that $f(x) \approx e^{\sqrt{A x}}$, because then the integral can be done by the saddlepoint method and agrees with $F(s)$. But can you suggest a method or reference to make this sort of statement precise? (Or is it wrong?)

In other words, can I reliably take the inverse Laplace transform of a function $F(s)$ that is known only approximately?

This question is motivated by a physics calculation, where $F(s)$ is the free energy at temperature $T=1/s$ and I am trying to infer something rigorous about the density of states $f(x)$.