find the limit $e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!}$

Question

I encountered the following problems when I was learning about the central limit theorem. Prove $$ e^{-n}\sum_{k=0}^{n}\frac{n^k}{k!}\rightarrow\frac{1}{2} \tag1\\ n\rightarrow\infty $$ This result can be easily obtained by using the central limit theorem of the Poisson distribution. But I don’t know how to use the following results to prove (1). $$ \sum_{\frac{n}{2}-x\sqrt{n}\leq k\leq \frac{n}{2}+x\sqrt{n}} \left( \begin{array}{c} n\\ k\\ \end{array} \right) \sim 2^n\frac{1}{\sqrt{2\pi}}\int_{-2x}^{2x}e^{-\frac{y^2}{2}}dy\tag2\\ n\rightarrow\infty\\ \forall x>0 $$ I use Stirling's formula to prove (2), but I still will not use (2) to prove (1). I am here to ask everyone for help.