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Showing $max(\frac{sin(x)}{x})=1$ is straight forward using l'hopital's rule. Is there another way to evaluate without using l'hopital's rule

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    $\begingroup$ It is not straightforward, it is circular. $\endgroup$ – DanielC Sep 18 '18 at 22:01
  • $\begingroup$ Do you mean Sup(sin x/x)? $\endgroup$ – gimusi Sep 18 '18 at 22:03
  • $\begingroup$ It's pretty easy using the Taylor series, but there is some sense it which is, as DanielC calls l'Hopital's rule, circular. $\endgroup$ – Acccumulation Sep 18 '18 at 22:19
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HINT

To prove that $\sup\left(\frac{\sin x}x\right)=1$ we can proceed showing that

  • $\frac{\sin x}x$ is even and not defined at $x=0$ then we can consider $x>0$
  • $\lim_{x\to 0} \frac{\sin x}x=1$
  • $g(x)=x-\sin x >0$ for $x>0$

Refer to the related

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