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Question Number 161248 by cortano last updated on 15/Dec/21

$$\underset{x\to 0}{\lim }\frac{{(1+\sin {}^{3}x)}^4-{(1+\tan {}^{3}x)}^4}{x^5}=?$$

Answered by bobhans last updated on 15/Dec/21

$$\mathcal{L}=2\times 2\times \underset{x\to 0}{\lim }\frac{\tan {}^3\left(\mathrm{x}\right)(\cos {}^3\mathrm{x}-1)}{{\mathrm{x}}^5}$$$$\mathcal{L}=12\times \underset{x\to 0}{\lim }\frac{\cos \mathrm{x}-1}{{\mathrm{x}}^2}$$$$\mathcal{L}=12\times \underset{x\to 0}{\lim }\frac{-2\sin {}^2\left(\frac{\mathrm{x}}{2}\right)}{{\mathrm{x}}^2}=-6$$

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