\(\lim\limits_{x\rightarrow-1}\dfrac{\left(x+1\right)^3}{x+3\sqrt[3]{x^2}+3\sqrt[3]{x}+1}\\ =\lim\limits_{x\rightarrow-1}\dfrac{\left(x+1\right)^3}{\left(\sqrt[3]{x}+1\right)^3}\\ =\lim\limits_{x\rightarrow-1}\dfrac{\left(\sqrt[3]{x}+1\right)^3\left(\sqrt[3]{x^2}-\sqrt[3]{x}+1\right)^3}{\left(\sqrt[3]{x}+1\right)^3}\\ =\lim\limits_{x\rightarrow-1}\left(\sqrt[3]{x^2}-\sqrt[3]{x}+1\right)^3=3^3=27\)