Is this a rigourous and correct way to evaluate a limit including an $n^{th}$ degree derivative? I'm a student in Physics' first year, I have an introductory knowledge on real analysis, and I'm not sure about how to solve the following limit:
$$\left.e^{2x}\frac{d^n}{dx^n} e^{-x^2}\right\rvert_{-\infty}^\infty$$
What I know---and can prove---is that, asymptotically and using small o notation,
$$\forall p,q,x\in\mathbb{R}:\ x^p = o_{x\to\infty}e^{qx}$$
and since $\frac{d^n}{dx^n} e^{-x^2} = P_n(x)e^{-x^2}$, with $P_n(x)$ an $n^{th}$ degree polynomial, it should follow that the above expression converges to zero due to the "faster" convergence of $e^{-x^2}$.
Now, is this correct? I'm maybe even more worried about the question "is this rigorous enough?" I don't like doing sloppy maths. I would appreciate any comment, advice or enlightening explanation on how to evaluate this expression rigorously.

The expression can be written
$$\frac{P(x)}{e^{(x-1)^2}}$$ or $$\frac{Q(x-1)}{e^{(x-1)^2}}$$
where $P,Q$ are polynomials.
Whatever their degree $d$, $$e^{(x-1)^2}=\Omega(|x-1|^{d+1}).$$
