calculus - $limlimits_{xtoinfty}left(cosleft(frac{1}{x}right)right)^x$

I want to find $\lim\limits_{x\to\infty}\left(\cos\left(\frac{1}{x}\right)\right)^{\!x}$.
I'd like to use the fact that
$$\lim_{x\to\infty} e^{\ln(\cos(1/x)^x)}$$
but I am not sure what to do after this.