\(\lim\limits_{x\rightarrow-2}\dfrac{\left(x+1\right)^2-2\left(x+1\right)-3}{\left(x+1\right)^3+2\left(x+1\right)^2-1}\)
\(=\lim\limits_{x\rightarrow-2}\dfrac{x^2+2x+1-2x-2-3}{x^3+3x^2+3x+1+2\left(x^2+2x+1\right)-1}\)
\(=\lim\limits_{x\rightarrow-2}\dfrac{x^2-4}{x^3+3x^2+3x+2x^2+4x+2}\)
\(=\lim\limits_{x\rightarrow-2}\dfrac{x^2-4}{x^3+5x^2+7x+2}\)
\(=\lim\limits_{x\rightarrow-2}\dfrac{\left(x-2\right)\left(x+2\right)}{x^3+2x^2+3x^2+6x+x+2}\)
\(=\lim\limits_{x\rightarrow-2}\dfrac{\left(x-2\right)\left(x+2\right)}{\left(x+2\right)\left(x^2+3x+1\right)}\)
\(=\lim\limits_{x\rightarrow-2}\dfrac{x-2}{x^2+3x+1}=\dfrac{-2-2}{\left(-2\right)^2+3\cdot\left(-2\right)+1}=\dfrac{-4}{4-6+1}=\dfrac{-4}{-1}=4\)
