\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt{1+x}-1}{\sqrt[3]{1+x}-1}\)
\(=\lim\limits_{x\rightarrow0}\left(\dfrac{1+x-1}{\sqrt{1+x}+1}:\dfrac{1+x-1}{\sqrt[3]{\left(1+x\right)^2}+\sqrt[3]{1+x}+1}\right)\)
\(=\lim\limits_{x\rightarrow0}\left(\dfrac{x}{\sqrt{1+x}+1}\cdot\dfrac{\sqrt[3]{\left(x+1\right)^2}+\sqrt[3]{x+1}+1}{x}\right)\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\sqrt[3]{\left(x+1\right)^2}+\sqrt[3]{x+1}+1}{\sqrt{x+1}+1}\)
\(=\dfrac{\sqrt[3]{\left(0+1\right)^2}+\sqrt[3]{0+1}+1}{\sqrt{0+1}+1}=\dfrac{1+1+1}{1+1}=\dfrac{3}{2}\)