Menu Close

lim-x-0-x-8-sin-8-x-x-10-




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

Leave a Reply

Your email address will not be published. Required fields are marked *