[tex]\lim_{x \to 1} \frac{\sqrt[3]{x}-1}{\sqrt{2+x}-\sqrt{3x}}[/tex]. Created by Nik133. algebra-ru

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Nik133 @Nik133

July 2022 1 27 Report

[tex]\lim_{x \to 1} \frac{\sqrt[3]{x}-1}{\sqrt{2+x}-\sqrt{3x}}[/tex]

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dtnth

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$\lim_{x \to 1} \frac{\sqrt[3]{x}-1}{\sqrt{2+x}-\sqrt{3x}}=\\ \lim_{x \to 1} \frac{(\sqrt[3]{x}-1)(\sqrt[3]{x^2}+\sqrt[3]{x}+1)(\sqrt{2+x}+\sqrt{3x})}{(\sqrt[3]{x^2}+\sqrt[3]{x}+1)(\sqrt{2+x}-\sqrt{3x})(\sqrt{2+x}+\sqrt{3x})}=\\ \lim_{x \to 1} \frac{(x-1)(\sqrt{2+x}+\sqrt{3x})}{(\sqrt[3]{x^2}+\sqrt[3]{x}+1)(2+x-3x)}=\\ \lim_{x \to 1} \frac{(x-1)(\sqrt{2+x}+\sqrt{3x})}{(\sqrt[3]{x^2}+\sqrt[3]{x}+1)(2-2x)}=\\ \lim_{x \to 1} \frac{(x-1)(\sqrt{2+x}+\sqrt{3x})}{-2(\sqrt[3]{x^2}+\sqrt[3]{x}+1)(x-1)}=\\$

$\lim_{x \to 1} \frac{\sqrt{2+x}+\sqrt{3x}}{-2(\sqrt[3]{x^2}+\sqrt[3]{x}+1)}=\\ \frac{\sqrt{2+1}+\sqrt{3*1}}{-2(\sqrt[3]{1^2}+\sqrt[3]{1}+1)}=\\ \frac{2\sqrt{3}}{-2*3}=\\ -\frac{\sqrt{3}}{3}$

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