\(\lim\limits\left(\dfrac{n+1}{n^2+2n}-\dfrac{1}{n-1}\right)\)
\(=\lim\limits\dfrac{\left(n+1\right)\left(n-1\right)-n^2-2n}{\left(n-1\right)\left(n^2+2n\right)}\)
\(=lim\dfrac{n^2-1-n^2-2n}{\left(n-1\right)\left(n^2+2n\right)}\)
\(=lim\dfrac{-2n-1}{n^3+n^2-2n}\)
\(=\lim\limits\dfrac{-\dfrac{2}{n^2}-\dfrac{1}{n^3}}{1+\dfrac{1}{n}-\dfrac{2}{n^2}}=0\)