<s>[INST] Whats the limit: $\lim\limits_{x \to 0} \frac{1-\sqrt[3]{\cos x}}{x(1-\cos\sqrt{x})}$ I am stuck at this one, i want to find it without using L'Hôpital's Rule.
$$\lim_{x \to 0} \frac{1-\sqrt[3]{\cos x}}{x(1-\cos\sqrt{x})}$$
 [/INST] $1-\cos \alpha=2\sin^2\frac{\alpha}2$,  $1-\cos \alpha\sim \frac {\alpha^2} 2(\alpha\to0)$, so
\begin{equation}
\begin{split}\lim_{x \to 0} \frac{1-\sqrt[3]{\cos x}}{x(1-\cos\sqrt{x})} &=\lim_{x \to 0} \frac{1-\cos x}{x(1-\cos\sqrt{x})(1+\sqrt[3]{\cos x}+\sqrt[3]{\cos^2 x})}\\&=\frac13 \lim_{x \to 0} \frac{1-\cos x}{x(1-\cos\sqrt{x})}\\
&=\frac13 \lim_{x \to 0} \frac{\frac {x^2} 2}{x\cdot\frac x2}\\
&=\frac13
\end{split}
\end{equation}
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