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Question Number 83287 by Power last updated on 29/Feb/20

Answered by mr W last updated on 29/Feb/20

sin x=x−(x^3 /(3!))+(x^5 /(5!))−..... ((sin x)/x)=1−(x^2 /(3!))+(x^4 /(5!))−..... sin ((π/2)×((sin x)/x))=sin ((π/2)−(π/2)((x^2 /(3!))−(x^4 /(5!))−.....)) =cos ((π/2)((x^2 /(3!))−(x^4 /(5!))−.....)) =1−(1/(2!))((π/2)((x^2 /(3!))))^2 +o(x^4 ) sin (((3π)/2)×((sin x)/x))=sin (((3π)/2)−((3π)/2)((x^2 /(3!))−(x^4 /(5!))−.....)) =−cos (((3π)/2)((x^2 /(3!))−(x^4 /(5!))−.....)) =−1+(1/(2!))(((3π)/2)((x^2 /(3!))))^2 +o(x^4 ) lim_(x→0) ((sin (((3π)/2)×((sin x)/x))+sin ((π/2)×((sin x)/x)))/(ax^n )) =lim_(x→0) (((1/(2!))(((3π)/2)((x^2 /(3!))))^2 −(1/(2!))((π/2)((x^2 /(3!))))^2 +o(x^4 ))/(ax^n )) =lim_(x→0) ((((π^2 x^4 )/(36))+o(x^4 ))/(ax^n ))=^(!) 1 ⇒n=4 ⇒a=(π^2 /(36))

$$\mathrm{sin}\:{x}={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−..... \\ $$$$\frac{\mathrm{sin}\:{x}}{{x}}=\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−..... \\ $$$$ \\ $$$$\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}×\frac{\mathrm{sin}\:{x}}{{x}}\right)=\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−\frac{\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}−\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−.....\right)\right) \\ $$$$=\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}−\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−.....\right)\right) \\ $$$$=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}!}\left(\frac{\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}\right)\right)^{\mathrm{2}} +{o}\left({x}^{\mathrm{4}} \right) \\ $$$$\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}×\frac{\mathrm{sin}\:{x}}{{x}}\right)=\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}−\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}−\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−.....\right)\right) \\ $$$$=−\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}−\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−.....\right)\right) \\ $$$$=−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}!}\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}\right)\right)^{\mathrm{2}} +{o}\left({x}^{\mathrm{4}} \right) \\ $$$$ \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\left(\frac{\mathrm{3}\pi}{\mathrm{2}}×\frac{\mathrm{sin}\:{x}}{{x}}\right)+\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}×\frac{\mathrm{sin}\:{x}}{{x}}\right)}{{ax}^{{n}} } \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\mathrm{1}}{\mathrm{2}!}\left(\frac{\mathrm{3}\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}\right)\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}!}\left(\frac{\pi}{\mathrm{2}}\left(\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}\right)\right)^{\mathrm{2}} +{o}\left({x}^{\mathrm{4}} \right)}{{ax}^{{n}} } \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\frac{\pi^{\mathrm{2}} {x}^{\mathrm{4}} }{\mathrm{36}}+{o}\left({x}^{\mathrm{4}} \right)}{{ax}^{{n}} }\overset{!} {=}\mathrm{1} \\ $$$$\Rightarrow{n}=\mathrm{4} \\ $$$$\Rightarrow{a}=\frac{\pi^{\mathrm{2}} }{\mathrm{36}} \\ $$

Commented by Power last updated on 01/Mar/20

thanks

$$\mathrm{thanks} \\ $$

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