<s>[INST] Limit of: $\lim_{x\to0}\frac{1-\sqrt[3]{\cos{x}}}{1-\cos{\sqrt[3]{x}}}$ I calculated this limit with L'Hospital's rule, but at the end it got rather complicated.
$\lim_{x\to0}\frac{1-\sqrt[3]{\cos{x}}}{1-\cos{\sqrt[3]{x}}}$
Is there some other more effective way for this limit?
 [/INST] Taylor expansions yield the result without much difficulty. We have
$$\sqrt[3]{1+y} = 1 + \frac{y}{3} + O(y^2),$$
and
$$\cos z = 1 - \frac{z^2}{2} + O(z^4),$$
hence
$$\sqrt[3]{\cos x} = \sqrt[3]{1 - x^2/2 + O(x^4)} = 1- \frac{x^2}{6} + O(x^4),$$
so
$$\frac{1-\sqrt[3]{\cos x}}{1-\cos \sqrt[3]{x}} = \frac{\frac{x^2}{6} + O(x^4)}{\frac{x^{2/3}}{2} + O(x^{4/3})} = \frac{x^{4/3}}{3} + O(x^2) \to 0.$$
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