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Question Number 74580 by chess1 last updated on 26/Nov/19

Answered by mind is power last updated on 26/Nov/19

$$\sqrt{1+\mathrm{t}}=1+\frac{\mathrm{t}}{2}+\mathrm{o}\left(\mathrm{t}\right)\Rightarrow \sqrt{{\mathrm{x}}^2+1}=1+\frac{{\mathrm{x}}^2}{2}+\mathrm{o}\left({\mathrm{x}}^2\right)$$$$\sqrt[3]{(\mathrm{t}+1)}=1+\frac{\mathrm{t}}{3}+\mathrm{o}\left(\mathrm{t}\right)\Rightarrow \sqrt[3]{1+{\mathrm{x}}^2}=1+\frac{{\mathrm{x}}^2}{3}+\mathrm{o}\left({\mathrm{x}}^2\right)$$$$\sqrt[4]{1+{\mathrm{x}}^4}=1+\frac{{\mathrm{x}}^4}{4}+0\left({\mathrm{x}}^4\right)$$$$\sqrt[5]{(1+{\mathrm{x}}^4)}=1+\frac{{\mathrm{x}}^4}{5}+0\left({\mathrm{x}}^4\right)$$$$\mathrm{We}\mathrm{get}\frac{\frac{{\mathrm{x}}^2}{6}+\mathrm{o}\left({\mathrm{x}}^2\right)}{\frac{{\mathrm{x}}^4}{20}+\mathrm{o}\left({\mathrm{x}}^4\right)}\sim \frac{10}{{3\mathrm{x}}^2}\to +\mathrm{\infty }$$

Commented by chess1 last updated on 27/Nov/19

$$\mathrm{thanks}$$

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