Can someone suggest how to solve this limit?

$$\lim_{x\to 0}\frac{\sin(\sin x)}{x}$$

If I substitute $y=\sin x$ then $\sin(\sin x)=\sin y$ while $x=\arcsin y$. Then the limit becomes $$\lim_{y\to 0}\frac{\sin y}{\arcsin y}$$

but this form is more complicated than the first one...

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    $\begingroup$ What is $senx$? Do you mean $\sin x$? $\endgroup$ – Omnomnomnom Nov 8 '17 at 16:51
  • $\begingroup$ Wonderful, I read this sen already a few times, thought I missed out on something. $\endgroup$ – Peter Szilas Nov 8 '17 at 16:59
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    $\begingroup$ @Omnomnomnom $\operatorname{sen}=\sin$ is a notation which, for some reason, is particularly common in Italian high-school books (it stands for "seno", which is the Italian name of the sine function). The odd fact is that, as far as I know, it completely disappears at university, so it remains like this teen-age crush you live with from 17 to 19, and then you never hear about it anymore. $\endgroup$ – user228113 Nov 8 '17 at 17:00
  • $\begingroup$ you're so right...i beg your pardon but I can't help to stop writing sen $\endgroup$ – Anne Nov 8 '17 at 17:03
  • $\begingroup$ @G.Sassatelli In addition to Italy, $\operatorname{sen}$ is used throughout Latin America, where the function also goes by the name seno. $\endgroup$ – gen-z ready to perish Nov 8 '17 at 17:11

Hint multiply top and bottom by $\sin x$ and break into a product of two limits.


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