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Question Number 139001 by bramlexs22 last updated on 21/Apr/21

$$\underset{x\to 0}{\lim }\frac{\sqrt[3]{1+\sin {}^2\mathrm{x}}-\sqrt[4]{1-2\tan \mathrm{x}}}{\sin \mathrm{x}+\tan {}^2\mathrm{x}}=?$$

Answered by EDWIN88 last updated on 21/Apr/21

$$\underset{x\to 0}{\lim }{\cos }^2\mathrm{x}\left(\frac{\sqrt[3]{1+\sin {}^2\mathrm{x}}-\sqrt[4]{1-2\tan \mathrm{x}}}{{\cos }^2\mathrm{x}\sin \mathrm{x}+\sin {}^2\mathrm{x}}\right)=$$$$\underset{x\to 0}{\lim }\frac{(1+\frac{{\mathrm{x}}^2}{3})-(1-\frac{2\mathrm{x}}{4})}{\mathrm{x}(1+\mathrm{x})}=$$$$\underset{x\to 0}{\lim }\frac{\mathrm{x}(\frac{\mathrm{x}}{3}+\frac{1}{2})}{\mathrm{x}(1+\mathrm{x})}=\frac{1}{2}$$

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