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Question Number 216698 by sniper237 last updated on 16/Feb/25

$$\underset{x\to +\mathrm{\infty }}{\lim }{}^3\sqrt{x+1}{-}^3\sqrt{x}\stackrel{?}{=}0$$

Answered by MrGaster last updated on 16/Feb/25

$$\sqrt[3]{x+1}-{}^3\sqrt{x}={(x+1)}^{\frac{1}{3}}-x^{\frac{1}{3}}$$$${(x+1)}^{\frac{1}{3}}=x^{\frac{1}{3}}+\frac{1}{3}{x}^{\frac{1}{3}-1}+O\left(x^{\frac{1}{3}-2}\right)$$$${(x+1)}^{\frac{1}{3}}-x^{\frac{1}{3}}=x^{\frac{1}{3}}+\frac{1}{3}{x}^{-\frac{2}{3}}+O\left(x^{-\frac{5}{3}}\right)-x^{\frac{1}{3}}$$$$\frac{1}{3}{x}^{-\frac{2}{3}}+O\left(x^{-\frac{5}{3}}\right)$$$$\underset{x\to {\mathrm{\infty }}^{+}}{\lim }(\frac{1}{3}{x}^{-\frac{2}{3}}+O\left(x^{-\frac{5}{3}}\right))=0$$

Answered by mehdee7396 last updated on 17/Feb/25

$$(\sqrt[3]{x+1}-\sqrt[3]{x})=\frac{1}{\sqrt[3]{{(x+1)}^2}+\sqrt[3]{(x+1)x}+\sqrt[3]{x^2}}$$$$\Rightarrow li\underset{x\to \mathrm{\infty }}{m}(\sqrt[3]{x+1}-\sqrt[3]{x})=0$$

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