How to show Poisson = Normal Distribution (in the right limit) without CLT

Question

If I take the Poisson PDF, $$\frac{e^{-\mu} \mu^x}{x!}$$ in the limit that $n$ is around the $\mu$ (i.e., not at the tail of the distribution) and $\mu$ is large this tends to the normal distribution, $$ \frac{1}{\sqrt{2\pi \mu}}e^{-(x-\mu)^2/2\mu} $$ This is really a statement about functional form so I suspect I should be able to show this without repeating the proof of the central limit theorem. Is there a way to show this using elementary Calculus (without resorting to the Central Limit Theorem)?