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Question Number 130176 by liberty last updated on 23/Jan/21

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{6\mathrm{x}+\sqrt{{8\mathrm{x}}^2-\sqrt{{2\mathrm{x}}^3+\sqrt{{4\mathrm{x}}^5}}}}{\sqrt[6]{{\mathrm{x}}^6-{\mathrm{x}}^3+\sqrt{{4\mathrm{x}}^5}}+\mathrm{x}}=?$$

Answered by EDWIN88 last updated on 23/Jan/21

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{6\mathrm{x}+\sqrt{{8\mathrm{x}}^2-\sqrt{{2\mathrm{x}}^3+{2\mathrm{x}}^2\sqrt{\mathrm{x}}}}}{\sqrt[6]{{\mathrm{x}}^6-{\mathrm{x}}^3+{2\mathrm{x}}^2\sqrt{\mathrm{x}}}+\mathrm{x}}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{6\mathrm{x}+\sqrt{{8\mathrm{x}}^2-\mathrm{x}\sqrt{2\mathrm{x}+2\sqrt{\mathrm{x}}}}}{\mathrm{x}\sqrt[6]{1-{\mathrm{x}}^{-3}+{2\mathrm{x}}^{-4}\sqrt{\mathrm{x}}}+\mathrm{x}}=$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{\mathrm{x}(6+\sqrt{8-{\mathrm{x}}^{-1}\sqrt{2\mathrm{x}+2\sqrt{\mathrm{x}}}})}{\mathrm{x}(\sqrt[6]{1-{\mathrm{x}}^{-3}+{2\mathrm{x}}^{-4}\sqrt{\mathrm{x}}}+1)}=$$$$\frac{6+\sqrt{8}}{2}=3+\sqrt{2}$$$$$$

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