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Question Number 214258 by efronzo1 last updated on 03/Dec/24

$$\underset{x\to \mathrm{\infty }}{\lim }\sqrt[6]{{\mathrm{x}}^6-{\mathrm{x}}^5}-\sqrt[6]{{\mathrm{x}}^6+{5\mathrm{x}}^5}=?$$

Answered by issac last updated on 03/Dec/24

$$-1$$

Answered by golsendro last updated on 03/Dec/24

$$=\underset{x\to 0}{\lim }\frac{\sqrt[6]{1-\mathrm{x}}-\sqrt[6]{1+5\mathrm{x}}}{\mathrm{x}}$$$$=\underset{x\to 0}{\lim }\frac{\sqrt{1-\mathrm{x}}-\sqrt{1+5\mathrm{x}}}{\mathrm{x}\left(\sqrt[3]{1-\mathrm{x}+\sqrt[3]{(1-\mathrm{x})(1+5\mathrm{x})}+\sqrt[3]{1+5\mathrm{x}}}\right)}$$$$=\underset{x\to 0}{\lim }\frac{\sqrt{1-\mathrm{x}}-\sqrt{1+5\mathrm{x}}}{3\mathrm{x}}$$$$=\underset{x\to 0}{\lim }\frac{-6\mathrm{x}}{3\mathrm{x}(\sqrt{1-\mathrm{x}}+\sqrt{1+5\mathrm{x}})}$$$$=-1$$

Answered by golsendro last updated on 03/Dec/24

$$=\underset{x\to 0}{\lim }\frac{\sqrt[6]{1-\mathrm{x}}-\sqrt[6]{1+5\mathrm{x}}}{\mathrm{x}}$$$$=\underset{x\to 0}{\lim }\frac{\sqrt{1-\mathrm{x}}-\sqrt{1+5\mathrm{x}}}{\mathrm{x}\left(\sqrt[3]{1-\mathrm{x}+\sqrt[3]{(1-\mathrm{x})(1+5\mathrm{x})}+\sqrt[3]{1+5\mathrm{x}}}\right)}$$$$=\underset{x\to 0}{\lim }\frac{\sqrt{1-\mathrm{x}}-\sqrt{1+5\mathrm{x}}}{3\mathrm{x}}$$$$=\underset{x\to 0}{\lim }\frac{-6\mathrm{x}}{3\mathrm{x}(\sqrt{1-\mathrm{x}}+\sqrt{1+5\mathrm{x}})}$$$$=-1$$

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