Why does $e^{-(x^2/2)} \approx \cos[\frac{x}{\sqrt{n}}]^n$ hold for large $n$? Why does this hold:
$$
e^{-x^2/2} = \lim_{n \to \infty} \cos^n \left( \frac{x}{\sqrt{n}} \right)
$$
I am not sure how to solve this using the limit theorem.

In a neighbourhood of the origin,
$$\log\cos z = -\frac{z^2}{2}\left(1+O(z^2)\right)\tag{1} $$
hence for any $x$ and for any $n$ big enough:
$$ \log\left(\cos^n\frac{x}{\sqrt{n}}\right)=-\frac{x^2}{2}\left(1+O\left(\frac{1}{n}\right)\right)\tag{2}$$
and the claim follows by exponentiating $(2)$:
$$ \cos^n\frac{x}{\sqrt{n}} = e^{-x^2/2}\cdot\left(1+O\left(\frac{1}{n}\right)\right).\tag{3}$$
