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Question Number 106072 by bemath last updated on 02/Aug/20

Answered by bobhans last updated on 02/Aug/20

$$\underset{x\to \pi /2}{\lim }\frac{\sqrt{2\sin {}^2\mathrm{x}+3\sin \mathrm{x}+4}}{\cot {}^2\mathrm{x}}=$$$$\mathrm{set}\mathrm{x}=\frac{\pi }{2}+\mathrm{h}\Rightarrow \underset{\mathrm{h}\to 0}{\lim }\frac{3}{\cot {}^2(\mathrm{h}+\frac{\pi }{2})}=$$$$\underset{\mathrm{h}\to 0}{\lim }\frac{3}{\tan {}^2\mathrm{h}}=\underset{\mathrm{h}\to 0}{\lim }\frac{{3\mathrm{h}}^2}{\tan {}^2\mathrm{h}}\times \underset{\mathrm{h}\to 0}{\lim }\frac{1}{{\mathrm{h}}^2}=\mathrm{\infty }$$$$$$$$$$

Answered by Dwaipayan Shikari last updated on 02/Aug/20

$$\sqrt{2{sin}^{2}x+3sinx+4}$$$$\sqrt{3}⩽\sqrt{2{sin}^{2}x+3sinx+4}⩽3$$$$\sqrt{2{sin}^{2}x+3sinx+4}\ne 0$$$$as{tan}^{2}x\to \mathrm{\infty }$$$$thewholefunctionreachestoinfinity$$

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