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Question Number 107163 by bobhans last updated on 09/Aug/20

Answered by john santu last updated on 09/Aug/20

$$⊵\mathrm{JS}⊴$$$$\underset{x\to 0}{\lim }\cos \mathrm{x}.\underset{x\to 0}{\lim }\frac{\mathrm{x}\sqrt{1+\frac{5}{\mathrm{x}}}}{2(\sqrt[5]{1-\frac{\sin \mathrm{x}}{8}}-1)}=$$$$1\times \underset{x\to 0}{\lim }\frac{\mathrm{x}\sqrt{1+\frac{5}{\mathrm{x}}}}{2(1-\frac{\sin \mathrm{x}}{40}-1)}=$$$$-20\times \underset{x\to 0}{\lim }\frac{\mathrm{x}\sqrt{1+\frac{5}{\mathrm{x}}}}{\sin \mathrm{x}}=-\mathrm{\infty }$$

Commented by bobhans last updated on 09/Aug/20

$$\mathrm{thank}\mathrm{you}$$

Answered by Dwaipayan Shikari last updated on 09/Aug/20

$$\underset{x\to 0}{\lim }\frac{\mathrm{cosx}\sqrt{{\mathrm{x}}^2+5\mathrm{x}}}{\sqrt[5]{32-4\mathrm{sinx}}-2}.\frac{32-4\mathrm{sinx}-32}{-4\mathrm{sinx}}$$$$\underset{x\to 0}{\lim }\frac{\sqrt{{\mathrm{x}}^2+5\mathrm{x}}}{-4\mathrm{sinx}}.5.2^4\to -\mathrm{\infty }$$

Commented by john santu last updated on 09/Aug/20

$$\mathrm{i}{\mathrm{don}}^{\prime }\mathrm{t}\mathrm{understand}\mathrm{your}\mathrm{process}$$$$\underset{x\to 0}{\lim }\frac{\cos \mathrm{x}\sqrt{{\mathrm{x}}^2+5\mathrm{x}}}{\sqrt[5]{32-4\sin \mathrm{x}}-2}.\frac{32-4\sin \mathrm{x}-32}{-4\sin \mathrm{x}}$$$$???\mathrm{what}\mathrm{formula}\mathrm{it}\mathrm{is}?$$

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