How do we reconcile the following difference? Specifically, what prevents us from changing the order of limits here?
$$\lim_{\lambda \to \infty} \sum_{n=0}^\infty e^{-\lambda} \lambda^n/n! = \lim_{\lambda \to \infty}1 = 1$$ $$\lim_{N \to \infty} \lim_{\lambda \to \infty} \sum_{n=0}^N e^{-\lambda} \lambda^n/n! = \lim_{N \to \infty}0 = 0$$
The polynomial dose not converges to $e^{\lambda}$ uniformly on the whole real line, so you cannot carelessly interchanging the limit and summation.
For a thorough discussion, see this link or any general textbook on analysis.