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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

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-.....$$$$\frac{\sin x}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-.....$$$$$$$$\sin \left(\frac{\pi }{2}\times \frac{\sin x}{x}\right)=\sin (\frac{\pi }{2}-\frac{\pi }{2}(\frac{x^2}{3!}-\frac{x^4}{5!}-.....))$$$$=\cos \left(\frac{\pi }{2}(\frac{x^2}{3!}-\frac{x^4}{5!}-.....)\right)$$$$=1-\frac{1}{2!}{\left(\frac{\pi }{2}\left(\frac{x^2}{3!}\right)\right)}^2+o\left(x^4\right)$$$$\sin \left(\frac{3\pi }{2}\times \frac{\sin x}{x}\right)=\sin (\frac{3\pi }{2}-\frac{3\pi }{2}(\frac{x^2}{3!}-\frac{x^4}{5!}-.....))$$$$=-\cos \left(\frac{3\pi }{2}(\frac{x^2}{3!}-\frac{x^4}{5!}-.....)\right)$$$$=-1+\frac{1}{2!}{\left(\frac{3\pi }{2}\left(\frac{x^2}{3!}\right)\right)}^2+o\left(x^4\right)$$$$$$$$\underset{x\to 0}{\lim }\frac{\sin \left(\frac{3\pi }{2}\times \frac{\sin x}{x}\right)+\sin \left(\frac{\pi }{2}\times \frac{\sin x}{x}\right)}{{ax}^n}$$$$=\underset{x\to 0}{\lim }\frac{\frac{1}{2!}{\left(\frac{3\pi }{2}\left(\frac{x^2}{3!}\right)\right)}^2-\frac{1}{2!}{\left(\frac{\pi }{2}\left(\frac{x^2}{3!}\right)\right)}^2+o\left(x^4\right)}{{ax}^n}$$$$=\underset{x\to 0}{\lim }\frac{\frac{{\pi }^2{x}^4}{36}+o\left(x^4\right)}{{ax}^n}\stackrel{!}{=}1$$$$\Rightarrow n=4$$$$\Rightarrow a=\frac{{\pi }^2}{36}$$

Commented by Power last updated on 01/Mar/20

$$\mathrm{thanks}$$

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