a: \(\lim\limits_{x\rightarrow-1}\dfrac{\sqrt[3]{x}-x}{x^2-x}\)
\(=\dfrac{\sqrt[3]{-1}-\left(-1\right)}{\left(-1\right)^2-\left(-1\right)}\)
\(=\dfrac{-1+1}{1+1}=\dfrac{0}{2}=0\)
b: \(\lim\limits_{x\rightarrow1}\dfrac{x^3-x^2-x+1}{x^3-3x+2}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x^3-x^2\right)-\left(x-1\right)}{x^3-x-2x+2}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{x^2\left(x-1\right)-\left(x-1\right)}{x\left(x^2-1\right)-2\left(x-1\right)}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x-1\right)\left(x^2-1\right)}{x\left(x-1\right)\left(x+1\right)-2\left(x-1\right)}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x-1\right)^2\cdot\left(x+1\right)}{\left(x-1\right)\left(x^2+x-2\right)}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x-1\right)\left(x+1\right)}{x^2+x-2}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x-1\right)\left(x+1\right)}{x^2+2x-x-2}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{\left(x-1\right)\left(x+1\right)}{\left(x+2\right)\left(x-1\right)}=\lim\limits_{x\rightarrow1}\dfrac{x+1}{x+2}=\dfrac{1+1}{1+2}=\dfrac{2}{3}\)
