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Question Number 169294 by cortano1 last updated on 28/Apr/22

Commented by qaz last updated on 28/Apr/22

$$\mathrm{Same}\mathrm{as}:\underset{\mathrm{x}\to 0}{\lim }\frac{\sin \tan \mathrm{x}-\tan \sin \mathrm{x}}{{\mathrm{x}}^7}=-\frac{1}{14}$$

Commented by cortano1 last updated on 28/Apr/22

$$youranswerwrong$$

Answered by bobhans last updated on 28/Apr/22

$$\underset{x\to 0}{\lim }\frac{\sinh \left(\sin x\right)-\sin \left(\sinh x\right)}{x^7}$$$$=\underset{x\to 0}{\lim }\frac{(x-\frac{1}{15}{x}^5+\frac{1}{90}{x}^7+O\left(x^9\right))-(x-\frac{1}{15}{x}^5-\frac{1}{90}{x}^7+O\left(x^9\right))}{x^7}$$$$=\underset{x\to 0}{\lim }\frac{\frac{1}{45}{x}^7+O\left(x^9\right)}{x^7}=\frac{1}{45}$$

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