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Question Number 116043 by bemath last updated on 30/Sep/20

$$\underset{x\to 0}{\lim }\frac{\sqrt{{\mathrm{x}}^2+{\mathrm{x}}^4}}{\mathrm{x}}?$$$$\int \sinh {}^2\left(\mathrm{x}\right)\cosh \left(\mathrm{x}\right)\mathrm{dx}$$$$\mathrm{find}\mathrm{x}\mathrm{from}\mathrm{equation}\cos \left({2\tan }^{-1}\left(\mathrm{x}\right)\right)=\frac{1}{2}$$

Answered by MJS_new last updated on 30/Sep/20

$$\frac{\sqrt{x^2+x^4}}{x}=\frac{x\sqrt{1+x^2}}{x}=\sqrt{1+x^2}\Rightarrow \mathrm{limit}\mathrm{is}1$$$$\int {\sinh }^{2}x\cosh xdx=\frac{1}{3}{\sinh }^{3}x+C$$$$\cos 2\arctan x=\frac{1-x^2}{1+x^2}=\frac{1}{2}\Rightarrow x=\pm \frac{\sqrt{3}}{3}$$

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