Question Number 197029 by cortano12 last updated on 06/Sep/23
$$\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{x}^{\mathrm{8}} −\mathrm{sin}\:^{\mathrm{8}} \mathrm{x}}{\mathrm{x}^{\mathrm{10}} }\:=? \\ $$
Answered by MM42 last updated on 06/Sep/23
$${lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\left({x}^{\mathrm{4}} +{sin}^{\mathrm{4}} {x}\right)\left({x}^{\mathrm{2}} +{sin}^{\mathrm{2}} {x}\right)\left({x}+{sinx}\right)\left({x}−{sinx}\right)}{{x}^{\mathrm{10}} } \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{2}{x}^{\mathrm{4}} ×\mathrm{2}{x}^{\mathrm{2}} ×\mathrm{2}{x}×\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} }{{x}^{\mathrm{10}} }\:=\frac{\mathrm{4}}{\mathrm{3}}\:\checkmark \\ $$$$ \\ $$