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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?

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This limit does not exist. You can simply take the particular subsequence: \begin{equation} x_n = \frac{1}{(2\pi n)^3} \end{equation}

and \begin{equation} x_n = \frac{1}{(\pi n)^3}, n\ \text{odd} \end{equation} so that they do not converge to the same value.

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