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Question Number 168143 by cortano1 last updated on 04/Apr/22

$$\underset{x\to 0}{\lim }\frac{1-\cos (1-\cos x)}{x^4}=?$$

Answered by qaz last updated on 04/Apr/22

$$$$$$\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos (1-\cos \mathrm{x})}{{\mathrm{x}}^4}=\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos (\frac{1}{2}{\mathrm{x}}^2+...)}{{\mathrm{x}}^4}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{\frac{1}{2}{(\frac{1}{2}{\mathrm{x}}^2+...)}^2}{{\mathrm{x}}^4}=\underset{\mathrm{x}\to 0}{\lim }\frac{\frac{1}{8}{\mathrm{x}}^4+\mathrm{o}\left({\mathrm{x}}^4\right)}{{\mathrm{x}}^4}=\frac{1}{8}$$

Answered by Mathspace last updated on 04/Apr/22

$$1-cosx\sim \frac{x^2}{2}\Rightarrow cos(1-cosx)\sim cos\left(\frac{x^2}{2}\right)$$$$\sim 1-\frac{x^4}{8}\Rightarrow 1-cos(1-cosx)\sim \frac{x^4}{8}$$$$\Rightarrow \frac{1-cos(1-cosx)}{x^4}\sim \frac{1}{8}\Rightarrow$$$${lim}_{x\to 0}\frac{1-cos(1-cosx)}{x^4}=\frac{1}{8}$$

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