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Question Number 112273 by bemath last updated on 07/Sep/20

$$\left(1\right)\underset{x\to \mathrm{\infty }}{\lim }{({\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}})}^{\frac{2}{\mathrm{x}}}=?$$$$\left(2\right)\underset{x\to \frac{\pi }{2}}{\lim }{\left(\cos \mathrm{x}\right)}^{-\mathrm{x}+\frac{\pi }{2}}?$$$$\left(3\right)\underset{x\to 0}{\lim }\sqrt{\frac{1+\tan \mathrm{x}}{{\mathrm{x}}^2}}-\sqrt{\frac{1-\sin \mathrm{x}}{{\mathrm{x}}^2}}?$$

Commented by Dwaipayan Shikari last updated on 07/Sep/20

$$e^{-2}{(1+e^{2x})}^{\frac{2}{x}}=y$$$$-2+\frac{2}{x}log(1+e^{2x})=logy$$$$-2+\frac{4e^{2x}}{1+e^{2x}}=logy$$$$-2+4(1-\frac{1}{1+e^{2x}})=logy$$$$-2+4=logy$$$$y=e^2$$

Commented by Dwaipayan Shikari last updated on 07/Sep/20

$$\underset{x\to \frac{\pi }{2}}{\lim }{\left(cosx\right)}^{-x+\frac{\pi }{2}}=y$$$$\Rightarrow -(x-\frac{\pi }{2})log\left(cosx\right)=logy$$$$\Rightarrow -(x-\frac{\pi }{2})(cosx-1)\frac{log(1+cosx-1)}{cosx-1}=logy$$$$\Rightarrow 2(x-\frac{\pi }{2}){sin}^2\frac{x}{2}=logy$$$$\Rightarrow logy=0$$$$y=1$$

Answered by bemath last updated on 07/Sep/20

Commented by mohammad17 last updated on 07/Sep/20

$$sirinstepfour\left(\frac{tanx+sinx}{\mid x\mid }\right)not\left(\frac{tanx-sinx}{\mid x\mid }\right)$$

Commented by bemath last updated on 07/Sep/20

$$\mathrm{oo}\mathrm{yes}\mathrm{you}\mathrm{are}\mathrm{right}$$

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