How does one show
$$\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}\left(1-\frac{t}{\sqrt{\alpha}}\right)^{-\alpha} = e^{t^2 / 2}?$$
Not homework, this is from this proof that the gamma distribution has a limiting distribution of the standard normal as $\alpha \to \infty$. It suggests using numerical techniques to find the limit of the above, but I would like to know if there is a good way to solve this manually. I tried L'Hopital's rule but got $e^t$, which is obviously incorrect.
Start with the exp/log trick, your limit is
$$\exp\left(\lim_{\alpha\to\infty}-t\sqrt\alpha+\alpha\log\left(1+{-t\over\sqrt\alpha}\right)\right).$$
Now let $\beta=\alpha^{-1}$.
$$\exp\left(\lim_{\beta\to 0^+}-t\beta^{-1/2}-\beta^{-1}\log\left(1+(-t\sqrt\beta)\right)\right).$$
Now here we can use the Maclaurin series for $\log(1+x)$ since we're going to $0$. We get
$$\exp\left(\lim_{\beta\to 0^+}-t\sqrt\beta^{-1/2}-\beta^{-1}(-t\sqrt\beta-{t^2\beta\over 2}+O(\beta^{3/2})\right).$$
The only term which doesn't vanish is ${t^2\over 2}$ so we recover the result.
