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Question Number 105114 by john santu last updated on 26/Jul/20

$$\underset{x\to 0}{\lim }\frac{2-\sqrt[3]{1-\sin {}^2\left(3x\right)}-\sqrt[3]{1-\sin {}^2\left(2x\right)}}{x^2}$$

Answered by bemath last updated on 26/Jul/20

$$\underset{x\to 0}{\lim }\frac{2-(1-\frac{\sin {}^2\left(3x\right)}{3})-(1-\frac{\sin {}^2\left(2x\right)}{3})}{x^2}$$$$\underset{x\to 0}{\lim }\frac{\frac{1}{3}(\sin {}^2\left(3x\right)+\sin {}^2\left(2x\right))}{x^2}=$$$$\to \frac{1}{3}(9+4)=\frac{13}{3}$$

Answered by OlafThorendsen last updated on 26/Jul/20

$$\underset{x\to 0}{\lim }\frac{2-\sqrt[3]{1-9x^2}-\sqrt[3]{1-4x^2}}{x^2}$$$$\underset{x\to 0}{\lim }\frac{2-(1-3x^2)-(1-\frac{4}{3}{x}^2)}{x^2}$$$$=\frac{13}{3}$$

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