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Question Number 216032 by efronzo1 last updated on 26/Jan/25

$$\underset{x\to 0}{\lim }\frac{1-\cos \mathrm{x}\sqrt{\cos 2\mathrm{x}}}{{\mathrm{x}}^2}=?$$

Answered by mr W last updated on 26/Jan/25

$$\cos x\sim 1-\frac{x^2}{2}$$$$\sqrt{\cos 2x}\sim 1-x^2$$$$\frac{1-\cos x\sqrt{\cos 2x}}{x^2}$$$$\sim \frac{1-(1-\frac{x^2}{2})(1-x^2)}{x^2}$$$$\sim \frac{(\frac{1}{2}+1)x^2}{x^2}$$$$\to \frac{3}{2}✓$$

Answered by golsendro last updated on 26/Jan/25

$$=\underset{x\to 0}{\lim }\frac{1-\cos \mathrm{x}}{{\mathrm{x}}^2}-\underset{x\to 0}{\lim }\left(\frac{\sqrt{\cos \mathrm{x}}-1}{{\mathrm{x}}^2}\right)\cos \mathrm{x}$$$$=\frac{1}{2}-\underset{x\to 0}{\lim }\frac{\cos \mathrm{x}-1}{{\mathrm{x}}^2(\sqrt{\cos \mathrm{x}}+1)}$$$$=\frac{1}{2}+1=\frac{3}{2}$$

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