# $\lim\limits_{x\to 0} \cos\left(\frac{1}{\sqrt[3]{x}}\right)$

I can't seem to find a way to solve $$\lim\limits_{x\to 0} \cos\left(\frac{1}{\sqrt[3]{x}}\right)$$ Is there really any way to solve it, or does it just not exist?

This limit does not exist. You can simply take the particular subsequence: $$x_n = \frac{1}{(2\pi n)^3}$$
and $$x_n = \frac{1}{(\pi n)^3}, n\ \text{odd}$$ so that they do not converge to the same value.