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Question Number 161409 by LEKOUMA last updated on 17/Dec/21

$$Calculate$$$$\underset{x\to 0}{\lim }\frac{1-\cos (1-\cos x)}{x^4}$$$$\underset{x\to 0}{\lim }\frac{1-\cos x\cos 2x\cos 3x}{x^2}$$

Commented by cortano last updated on 17/Dec/21

$$\left(2\right)\underset{x\to 0}{\lim }\frac{1-\cos x\cos 2x\cos 3x}{x^2}$$$$=\underset{x\to 0}{\lim }\frac{1-\cos x+\cos x-\cos x\cos 2x\cos 3x}{x^2}$$$$=\underset{x\to 0}{\lim }\frac{1-\cos x}{x^2}+\underset{x\to 0}{\lim }\frac{\cos x(1-\cos 2x\cos 3x)}{x^2}$$$$=\frac{1}{2}+\underset{x\to 0}{\lim }\frac{1-\cos 2x+\cos 2x-\cos 2x\cos 3x}{x^2}$$$$=\frac{1}{2}+\underset{x\to 0}{\lim }\frac{1-\cos 2x}{x^2}+\underset{x\to 0}{\lim }\frac{\cos 2x(1-\cos 3x)}{x^2}$$$$=\frac{1}{2}+2+\underset{x\to 0}{\lim }\frac{1-\cos 3x}{x^2}$$$$=\frac{5}{2}+\frac{9}{2}=\frac{14}{2}=7$$

Answered by qaz last updated on 17/Dec/21

$$\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos \mathrm{xcos}2\mathrm{xcos}3\mathrm{x}}{{\mathrm{x}}^2}$$$$=-\underset{\mathrm{x}\to 0}{\lim }\frac{\ln \left(\cos \mathrm{xcos}2\mathrm{xcos}3\mathrm{x}\right)}{{\mathrm{x}}^2}$$$$=-\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{lncos}\mathrm{x}}{{\mathrm{x}}^2}-\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{lncos}2\mathrm{x}}{{\mathrm{x}}^2}-\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{lncos}3\mathrm{x}}{{\mathrm{x}}^2}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos \mathrm{x}}{{\mathrm{x}}^2}+\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos 2\mathrm{x}}{{\mathrm{x}}^2}+\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos 3\mathrm{x}}{{\mathrm{x}}^2}$$$$=\frac{1}{2}+\frac{1}{2}\cdot 2^2+\frac{1}{2}\cdot 3^2$$$$=7$$$$------------$$$$\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos (1-\cos \mathrm{x})}{{\mathrm{x}}^4}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{\frac{1}{2}{(1-\cos \mathrm{x})}^2}{{\mathrm{x}}^4}$$$$=\frac{1}{2}\underset{\mathrm{x}\to 0}{\lim }\frac{{\left(\frac{1}{2}{\mathrm{x}}^2\right)}^2}{{\mathrm{x}}^4}$$$$=\frac{1}{8}$$

Commented by LEKOUMA last updated on 17/Dec/21

$$Good!Manythank$$

Answered by cortano last updated on 17/Dec/21

$$\left(1\right)\underset{x\to 0}{\lim }\frac{1-\cos (1-\cos x)}{x^4}$$$$=\underset{x\to 0}{\lim }\frac{2\sin {}^2\left(\frac{1-\cos x}{2}\right)}{x^4}$$$$=\underset{x\to 0}{\lim }\frac{2\sin {}^2\left(\frac{2\sin {}^2\left(\frac{x}{2}\right)}{2}\right)}{x^4}$$$$=2\times \frac{4\times \frac{1}{16}}{4}=2\times \frac{1}{16}=\frac{1}{8}$$

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