Existence of $A$ such that $ \lim_{x\to\infty}\operatorname{poly}(x) e^{-x} \sum_{n\in A} \frac{x^n}{n!}=1 $

I want to know if there exists a set $A \subseteq \mathbb{N}$ such that$$\lim_{x\to\infty} x^2 e^{-x} \sum_{n\in A} \dfrac{x^n}{n!}=1$$

More generally, the question will be the existence of a set $A$ that$$\lim_{x\to\infty}\operatorname{poly}(x) e^{-x} \sum_{n\in A} \dfrac{x^n}{n!}=1$$

When $A$ is finite, it is obvious that the limit must be $0$. But when $A$ is infinite, the structure of $A$ can be very complex, and I don't know how to proceed further.