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Question Number 184795 by Spillover last updated on 11/Jan/23

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

Commented by MJS_new last updated on 11/Jan/23

$$\frac{3}{2}$$

Answered by cortano1 last updated on 12/Jan/23

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

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