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Question Number 171079 by greougoury555 last updated on 07/Jun/22

$$\underset{x\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+\sqrt{1-x}}}-\sqrt{1+\sqrt{1+\sqrt{1+x}}}}{x}=?$$

Commented by benhamimed last updated on 07/Jun/22

$$\frac{-1}{24}$$

Commented by greougoury555 last updated on 08/Jun/22

Answered by qaz last updated on 07/Jun/22

$$\underset{\mathrm{x}\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+\sqrt{1-\mathrm{x}}}}-\sqrt{1+\sqrt{1+\sqrt{1+\mathrm{x}}}}}{\mathrm{x}}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{1}{\mathrm{x}}(\sqrt{1+\sqrt{2-\frac{1}{2}\mathrm{x}+\mathrm{o}\left(\mathrm{x}\right)}}-\sqrt{1+\sqrt{2+\frac{1}{2}\mathrm{x}+\mathrm{o}\left(\mathrm{x}\right)}})$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{1}{\mathrm{x}}(\sqrt{1+\sqrt{2}(1-\frac{1}{8}\mathrm{x}+\mathrm{o}\left(\mathrm{x}\right))}-\sqrt{1+\sqrt{2}(1+\frac{1}{8}\mathrm{x}+\mathrm{o}\left(\mathrm{x}\right))})$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{\sqrt{1+\sqrt{2}}}{\mathrm{x}}(\sqrt{1-\frac{\sqrt{2}}{8(1+\sqrt{2})}\mathrm{x}+\mathrm{o}\left(\mathrm{x}\right)}-\sqrt{1+\frac{\sqrt{2}}{8(1+\sqrt{2})}\mathrm{x}+\mathrm{o}\left(\mathrm{x}\right)})$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{\sqrt{1+\sqrt{2}}}{\mathrm{x}}(-\frac{\sqrt{2}}{8(1+\sqrt{2})}\mathrm{x}+\mathrm{o}\left(\mathrm{x}\right))$$$$=-\frac{1}{8}\sqrt{\frac{2}{1+\sqrt{2}}}$$

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