I am working on an MGF question similar to https://math.stackexchange.com/posts/2269257 but I was specifically looking for help on the part they leave out, evaluating a limit like the following as $n \to \infty$
$$M_{Y_n}(t) = e^{-t\sqrt{n}} \left(\frac{1}{1 - t/\sqrt{n}}\right)^n.$$
Hint
The log of this is $$ -t\sqrt{n} - n\log(1-t/\sqrt n)$$ and the limit can be found by Taylor expanding the log.