Sử dụng hằng đẳng thức: $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\ldots+ab^{n-2}+b^{n-1}$
Ta sẽ có: $\lim\limits_{x\to 1}\dfrac{x^n-1}{x-1}=\lim\limits_{x\to 1}\dfrac{(x-1)(x^{n-1}+x^{n-2}+\ldots+x+1}{x-1}=n$
Từ đó:
\begin{align*}
&\lim\limits_{x\to 1}\dfrac{x^{2016}-2016x+2015}{(x-1)^2}\\
=&\lim\limits_{x\to 1}\dfrac{x^{2016}-1-2016(x-1)}{x-1}\\
=&\lim\limits_{x\to 0}\dfrac{x^{2015}+x^{2014}+\ldots +x-2015}{x-1}\\
=&\lim\limits_{x\to 0}\left(\dfrac{x^{2015}-1}{x-1}+\dfrac{x^{2014}-1}{x-1}+\ldots+\dfrac{x-1}{x-1}\right)\\
=&2015+2014+\ldots+1=\dfrac{2015.2016}{2}=2031120
\end{align*}