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Question Number 216055 by efronzo1 last updated on 26/Jan/25

$$\underset{x\to 0}{\lim }\frac{\sin {}^{2}2\mathrm{x}}{\sqrt[3]{\cos \mathrm{x}}-\sqrt[4]{\cos \mathrm{x}}}=?$$

Answered by golsendro last updated on 27/Jan/25

$$=\underset{x\to 0}{\lim }\frac{4\cos {}^2\mathrm{x}(1-\cos {}^2\mathrm{x})}{\sqrt[3]{\cos \mathrm{x}}-\sqrt[4]{\cos \mathrm{x}}}$$$$=8.\underset{x\to 0}{\lim }\frac{1-\cos \mathrm{x}}{\sqrt[3]{\cos \mathrm{x}}-\sqrt[4]{\cos \mathrm{x}}}$$$$\mathrm{Let}\cos \mathrm{x}={\mathrm{t}}^{12}$$$$=8\underset{\mathrm{t}\to 1}{\lim }\frac{1-{\mathrm{t}}^{12}}{{\mathrm{t}}^4-{\mathrm{t}}^3}$$$$=8\underset{\mathrm{t}\to 1}{\lim }\frac{(1-\mathrm{t})({\mathrm{t}}^{11}+{\mathrm{t}}^{10}+...+1)}{{\mathrm{t}}^3(\mathrm{t}-1)}$$$$=-8.\left(12\right)=-96$$

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