Question about limit $limlimits_{nrightarrowinfty}cos^nleft(frac{omega}{sqrt{n}}right)$

My question is how to calculate this limit. $$\lim_{n\rightarrow\infty}\cos^n\left(\frac{\omega}{\sqrt{n}}\right)$$ I know that the answer is $e^{-\frac{\omega^2}{2}}$.

Attempts: I tried to reduce the limit to the known limit $$\lim_{n\rightarrow \infty}\left(1+\frac{a}{n}\right)^n=e^{a}$$ So, I wrote $$\cos\left(\frac{\omega}{\sqrt{n}}\right)\approx1-\frac{\omega^2}{2n}$$ using the cosine Taylor series, and stop there because $\frac{\omega}{\sqrt{n}}$ gets very small as $n\rightarrow \infty$. Then, the limit is $$\lim_{n\rightarrow\infty}\cos^n\left(\frac{\omega}{\sqrt{n}}\right)=\lim_{n\rightarrow \infty}\left(1-\frac{\omega^2}{2n}\right)^n=e^{-\frac{\omega^2}{2}}$$

I also tried using $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ and then using the binomial theorem with no success.

Is this answer correct?