$\mathop {\lim }\limits_{x \to 0}\frac{\sqrt{x^3+1}-\left ( x^2+1 \right )^{\frac{1}{3}}}{x^3}$
$=\mathop {\lim }\limits_{x \to 0}\frac{\sqrt{x^3+1}-1+1-\sqrt[3]{x^2+1}}{x^3}$
$=\mathop {\lim }\limits_{x \to 0}\frac{\frac{x^3}{\sqrt{x^3+1}+1}-\frac{x^2}{1+\sqrt[3]{x^2+1}+\sqrt[3]{(x^2+1)^2}}}{x^3}$$=\mathop {\lim }\limits_{x \to 0}\frac{\frac{x}{\sqrt{x^3+1}+1}-\frac{1}{1+\sqrt[3]{x^2+1}+\sqrt[3]{(x^2+1)^2}}}{x}$
$=- \infty$
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