Задача №1953

\[ \lim_{x\to{1}}\left(\frac{m}{1-x^m}-\frac{n}{1-x^n}\right) =\lim_{x\to{1}}\frac{m-mx^n-n+nx^m}{\left(1-x^m\right)\left(1-x^n\right)} =\lim_{t\to{0}}\frac{m-m(1+t)^n-n+n(1+t)^m}{\left(1-(1+t)^m\right)\left(1-(1+t)^n\right)}=\\ =\lim_{t\to{0}}\frac{m-m\left(1+nt+\frac{(n-1)n}{2}t^2+o\left(t^2\right) \right)-n+n\left(1+mt+\frac{(m-1)m}{2}t^2+o\left(t^2\right)\right)}{\left(1-(1+mt+o(t))\right)\left(1-(1+nt+o(t))\right)}=\\ =\lim_{t\to{0}}\frac{\frac{-mn(n-1)}{2}+\frac{(m-1)nm}{2}+\frac{o\left(t^2\right)}{t^2}}{\left(-m+\frac{o(t)}{t}\right)\cdot\left(-n+\frac{o(t)}{t}\right)} =\frac{m-n}{2}. \]