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Question Number 122338 by benjo_mathlover last updated on 15/Nov/20

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

Commented by liberty last updated on 16/Nov/20

$$\mathrm{Find}\underset{x\to \mathrm{\infty }}{\lim }\frac{3\sqrt{\mathrm{x}}+3\sqrt[3]{\mathrm{x}}+5\sqrt[5]{\mathrm{x}}}{\sqrt{2\mathrm{x}-3}+\sqrt[3]{3\mathrm{x}-2}}.$$$$\mathrm{Solution}:$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{3+\frac{3}{\sqrt[6]{\mathrm{x}}}+\frac{5}{\sqrt[10]{{\mathrm{x}}^3}}}{\sqrt{2-\frac{3}{\mathrm{x}}}+\frac{\sqrt[6]{{(3\mathrm{x}-2)}^2}}{\sqrt[6]{{\mathrm{x}}^3}}}$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{3+\frac{3}{\sqrt[6]{\mathrm{x}}}+\frac{5}{\sqrt[10]{{\mathrm{x}}^3}}}{\sqrt{2-\frac{3}{\mathrm{x}}}+\sqrt[6]{\frac{{9\mathrm{x}}^2-12\mathrm{x}+4}{{\mathrm{x}}^3}}}$$$$\underset{x\to \mathrm{\infty }}{\lim }\frac{3+\frac{3}{\sqrt[6]{\mathrm{x}}}+\frac{5}{\sqrt[10]{\mathrm{x}}}}{\sqrt{2-\frac{3}{\mathrm{x}}}+\sqrt[6]{\frac{9}{\mathrm{x}}-\frac{12}{{\mathrm{x}}^2}+\frac{4}{{\mathrm{x}}^3}}}=\frac{3}{\sqrt{2}}.▴$$

Commented by benjo_mathlover last updated on 16/Nov/20

$$yes...thanks$$

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