Question Number 128745 by shaker last updated on 10/Jan/21
Answered by liberty last updated on 10/Jan/21
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{1}+\mathrm{mnx}+\frac{\mathrm{n}\left(\mathrm{n}−\mathrm{1}\right)}{\mathrm{2}}\mathrm{m}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \right)−\left(\mathrm{1}+\mathrm{mnx}+\frac{\mathrm{m}\left(\mathrm{m}−\mathrm{1}\right)}{\mathrm{2}}\mathrm{n}^{\mathrm{2}} \mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} }= \\ $$$$\frac{\mathrm{m}^{\mathrm{2}} \left(\mathrm{n}^{\mathrm{2}} −\mathrm{1}\right)−\mathrm{n}^{\mathrm{2}} \left(\mathrm{m}^{\mathrm{2}} −\mathrm{1}\right)}{\mathrm{2}}=\frac{\mathrm{n}^{\mathrm{2}} −\mathrm{m}^{\mathrm{2}} }{\mathrm{2}} \\ $$