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Question Number 190435 by mustafazaheen last updated on 02/Apr/23
  how is solution  lim_(x→0) ((x^(10) ∙sin^4 x∙cos^8 x∙(x+1)^3 )/(x^4 +3x^3 +3x^2 +x))=?
$$ \\ $$$$\mathrm{how}\:\mathrm{is}\:\mathrm{solution} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{10}} \centerdot\mathrm{sin}^{\mathrm{4}} \mathrm{x}\centerdot\mathrm{cos}^{\mathrm{8}} \mathrm{x}\centerdot\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{x}^{\mathrm{4}} +\mathrm{3x}^{\mathrm{3}} +\mathrm{3x}^{\mathrm{2}} +\mathrm{x}}=? \\ $$$$ \\ $$
Answered by JDamian last updated on 02/Apr/23
0
$$\mathrm{0} \\ $$
Commented by mustafazaheen last updated on 03/Apr/23
how is solution
$$\mathrm{how}\:\mathrm{is}\:\mathrm{solution} \\ $$
Answered by mr W last updated on 03/Apr/23
=lim_(x→0) ((x^(10) ×sin^4  x×cos^8  x×(x+1)^3 )/(x(1+3x+3x^2 +x^3 )))  =lim_(x→0) ((x^9 ×sin^4  x×cos^8  x×(x+1)^3 )/(1+3x+3x^2 +x^3 ))  =((0^9 ×0^4 ×1^8 ×1^3 )/1)  =0
$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}^{\mathrm{10}} ×\mathrm{sin}^{\mathrm{4}} \:{x}×\mathrm{cos}^{\mathrm{8}} \:{x}×\left({x}+\mathrm{1}\right)^{\mathrm{3}} }{{x}\left(\mathrm{1}+\mathrm{3}{x}+\mathrm{3}{x}^{\mathrm{2}} +{x}^{\mathrm{3}} \right)} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}^{\mathrm{9}} ×\mathrm{sin}^{\mathrm{4}} \:{x}×\mathrm{cos}^{\mathrm{8}} \:{x}×\left({x}+\mathrm{1}\right)^{\mathrm{3}} }{\mathrm{1}+\mathrm{3}{x}+\mathrm{3}{x}^{\mathrm{2}} +{x}^{\mathrm{3}} } \\ $$$$=\frac{\mathrm{0}^{\mathrm{9}} ×\mathrm{0}^{\mathrm{4}} ×\mathrm{1}^{\mathrm{8}} ×\mathrm{1}^{\mathrm{3}} }{\mathrm{1}} \\ $$$$=\mathrm{0} \\ $$

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