In order to obtain an estimate I'm wondering if there is a positive function $f:\mathbb{R}\to \mathbb{R}_+$ such that $f(x) < |x|$ for $|x|$ large enough (say $|x|\ge C_0>0$) so that the following inequality holds whenever $|x|\ge C_0$: $$ e^{f(x)}\le C_1^{N+1}(f(x))^{N+1}\sqrt{|x|}(1 + |x|)^{-1/2-N} $$ for every $N\in \mathbb{N}_0$ and some constant $C_1>0$?
I've tried many candidates such as $$ f(x) = C(1 + |x|)\log (1 + \frac{1}{|x|}) $$ but can't find anything that works. (And I don't know for sure whether such a function exists.)
This is only an idea.
Take limits. For every positive function $f(x)$, the exponential growth of $e^{f(x)}$ will always overcome the polynomial growth of $({C_1f(x))}^{N + 1}$.
The question is whether or not can $\frac{ \sqrt{|x|} }{ {(1 + |x|)}^{N + \frac12} }$ compensate the growth of the left side.
You have lower bound for $f(x) \lt |x|$. Try using it.