Lời giải:\(\lim\limits_{x\to 1}\left(\frac{m}{1-x^m}-\frac{n}{1-x^n}\right)=\lim\limits_{x\to 1}\left(\frac{m}{1-x^m}-\frac{1}{1-x}-\frac{n}{1-x^n}+\frac{1}{1-x}\right)\)
\(=\lim\limits_{x\to 1}\left[\frac{m-(1+x+...+x^{m-1})}{1-x^m}-\frac{n-(1+x+..+x^{n-1})}{1-x^n}\right]\)
\(=\lim\limits_{x\to 1}\left[\frac{(1-x)+(1-x^2)+...+(1-x^{m-1})}{1-x^m}-\frac{(1-x)+(1-x^2)+...+(1-x^{n-1})}{1-x^n}\right]\)
\(\lim\limits_{x\to 1}\left[\frac{1+(x+1)+...+(1+x+...x^{m-2})}{1+x+...+x^{m-1}}-\frac{1+(1+x)+...+(1+x+...+x^{n-2})}{1+x+...x^{n-1}}\right]\)
\(=\frac{m(m-1)}{2m}-\frac{n(n-1)}{2n}=\frac{m-1}{2}-\frac{n-1}{2}=\frac{m-n}{2}\)
C2: Xài L'Hospital
\(\lim\limits_{x\rightarrow1}\dfrac{m-m.x^n-n+n.x^m}{1-x^m-x^n+x^{m+n}}=\lim\limits_{x\rightarrow1}\dfrac{m.n.x^{m-1}-m.n.x^{n-1}}{\left(m+n\right)x^{m+n-1}-m.x^{m-1}-n.x^{n-1}}\)
\(=\lim\limits_{x\rightarrow1}\dfrac{m.n.\left(m-1\right).x^{m-2}-m.n.\left(n-1\right).x^{n-2}}{\left(m+n\right).\left(m+n-1\right)x^{m+n-2}-m\left(m-1\right)x^{m-2}-n\left(n-1\right)x^{n-2}}\)
\(=\dfrac{m^2n-mn-mn^2+mn}{m^2+2mn-m+n^2-n-m^2+m-n^2+n}=\dfrac{mn\left(m-n\right)}{2mn}=\dfrac{m-n}{2}\)
