\(\lim\limits_{x\rightarrow1}\frac{x^{n+1}-\left(n+1\right)x+n}{\left(x-1\right)^2}=\lim\limits_{x\rightarrow1}\frac{\left(n+1\right)x^n-\left(n+1\right)}{2\left(x-1\right)}=\lim\limits_{x\rightarrow1}\frac{n\left(n+1\right)x^{n-1}}{2}=\frac{n\left(n+1\right)}{2}\)
Phức tạp hơn thì làm như sau:
\(\lim\limits_{x\rightarrow1}\frac{x\left(x^n-nx+n-1\right)+n\left(x^2-2x+1\right)}{\left(x-1\right)^2}\)
\(=\lim\limits_{x\rightarrow1}\frac{x\left(x-1\right)^2\left(x^{n-1}+2x^{n-2}+...+\left(n-2\right)x+n-1\right)+n\left(x-1\right)^2}{\left(x-1\right)^2}\)
\(=\lim\limits_{x\rightarrow1}\left[x\left(x^{n-1}+2x^{n-2}+...+n-1\right)+n\right]\)
\(=\left(1+2+...+n-1+n\right)=\frac{n\left(n+1\right)}{2}\)
