با استفاده از اتحاد $(a-b)({{a}^{2}}+ab+{{b}^{2}})={{a}^{3}}-{{b}^{3}}$ خواهیم داشت: $\underset{x\to +\infty }{\mathop{\lim }}\,{{x}^{2}}(x-\sqrt[3]{{{x}^{3}}+1})$ $=\underset{x\to +\infty }{\mathop{\lim }}\,{{x}^{2}}(x-\sqrt[3]{{{x}^{3}}+1})\times \frac{{{x}^{2}}+x\sqrt[3]{{{x}^{3}}+1}+\sqrt[3]{{{({{x}^{3}}+1)}^{2}}}}{{{x}^{2}}+x\sqrt[3]{{{x}^{3}}+1}+x\sqrt[3]{{{({{x}^{3}}+1)}^{2}}}}$ $=\underset{x\to +\infty }{\mathop{\lim }}\,\frac{{{x}^{2}}({{x}^{3}}-{{x}^{3}}-1)}{{{x}^{2}}+x\sqrt[3]{{{x}^{3}}+1}+\sqrt[3]{{{({{x}^{3}}+1)}^{2}}}}$ $=\underset{x\to +\infty }{\mathop{\lim }}\,\frac{-{{x}^{2}}}{{{x}^{2}}+{{x}^{2}}+{{x}^{2}}}=\underset{x\to +\infty }{\mathop{\lim }}\,\frac{-{{x}^{2}}}{3{{x}^{2}}}=-\frac{1}{3}$