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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1It is not straightforward, it is circular. – DanielC Sep 18 '18 at 22:01
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Do you mean Sup(sin x/x)? – user Sep 18 '18 at 22:03
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It's pretty easy using the Taylor series, but there is some sense it which is, as DanielC calls l'Hopital's rule, circular. – 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 $x>0$
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