real analysis - Show that $lim_{n rightarrow infty} frac{sin^{n}(frac{x}{sqrt{n}})}{left(frac{x}{sqrt{n}} right)^n} = e^{-frac{x^2}{6}} $

I am wondering about a limit that wolframalpha got me and that you can find here wolframalpha

It says that $$\lim_{n \rightarrow \infty} \frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} \right)^n} = e^{-\frac{x^2}{6}} $$

Does anybody know if there is a "easy" way to get this?

Answer

As
$$
1 - \frac{\sin t}t \sim_{t\to 0} \frac{t^2}{3!}

\\
\log (1+\epsilon) \sim_{\epsilon\to 0} \epsilon
$$

you have
$$
\frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} \right)^n}
= \exp \left[
n\log \frac{\sin \frac x{\sqrt n}}{\frac x{\sqrt n}}
\right]

\to \exp \left[
n\left( -\frac 16 \left(\frac x{\sqrt n}\right)^2 \right)
\right]
= \exp\left( -\frac{x^2}6\right)
$$