Suppose we have a function defined by $$\varphi(s)=\int_{-\infty}^\infty f(x,s)\,dx$$ defined for $s\in S\subseteq \mathbb{R}$. Suppose we know that it blows up at $a\in \partial S$, and we want to figure out the order of the function $\varphi$ as $s\to a$. How can we do that in general?
As a concrete example, there is a trick to determine that $$\int_{-\infty}^\infty e^{-sx^2}\,dx=\sqrt{\frac{\pi}{s}}.$$ But suppose we didn't know that. How could we show that $\int_{-\infty}^\infty e^{-sx^2}\,dx$ was $O(s^{-\frac{1}{2}})$ at $s=0$?
Ideas: Taylor series, then integrating term by term, justifying with additional assumptions if necessary?