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Question Number 187645 by Mingma last updated on 19/Feb/23

Answered by Ar Brandon last updated on 19/Feb/23

$$\mathcal{L}=\underset{x\to 0}{\lim }\frac{{\sin }^4\left(\pi \cos x\right)}{1-\cos (1-\cos (1-\cos x))}$$$$=\underset{x\to 0}{\lim }\frac{{\sin }^4(\pi -\frac{\pi x^2}{2})}{\frac{x^8}{128}}=\underset{x\to 0}{\lim }\frac{{\sin }^4\left(\frac{\pi x^2}{2}\right)}{\frac{x^8}{128}}$$$$=\underset{x\to 0}{\lim }{\left(\frac{\sin \left(\frac{\pi x^2}{2}\right)}{\frac{\pi x^2}{2}}\right)}^4\times 8{\pi }^4=8{\pi }^4$$

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