Asymptotic behaviour of $\int_0^1 g(x)\exp(-nx)dx$ as $n\rightarrow\infty$

Question

Let $g:(0,1]\rightarrow\mathbb{R}_+$ be an invertible monotonically non-increasing function that integrates to $1$ and has $g(1)=0$, $g(0)=\infty$; eg. $g(x)=x^{-1/2}-1$ or $g(x)=\ln(1/x)$. I believe it is the case that: $$ \int_0^1 g(x)\exp(-nx)dx=\Theta(g(1/n)/n) $$ but I'm not having any luck showing this in general. I'd also like to know if there are particular tools that work well for computing integrals of this form generally.