[tex]\lim\limits_{x\to 1}\dfrac{nx(x^m-1)-mx(x^n-1)}{(x-1)^2}=\\ \\ = \lim\limits_{x\to 1}\dfrac{nx^{m+1}-nx-mx^{n+1}+mx}{(x-1)^2}= \\ \\ = \lim\limits_{x\to 1}\dfrac{n(m+1)x^{m}-n-m(n+1)x^{n}+m}{2(x-1)}= \\ \\ = \lim\limits_{x\to 1}\dfrac{n(m+1)mx^{m-1}-m(n+1)nx^{n-1}}{2} = \\ \\ = \dfrac{mn(m+1)-mn(n+1)}{2} = \boxed{\dfrac{mn(m-n)}{2}}[/tex]
