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we gets (sinx)'=cosx from lim x→0 sin(x)/x=1

Geometric proof of it is based on the fact that the area of sector of unit circle with central angle x is x/2

However, when we calculate the area of unit circle, we use the integral of trigonometrical function which derive from (sinx)'=cosx. That is an circulation logic!

Also, area formula of polar coordination also derives from area of sector, so we can't use it to proof lim x→0 sin(x)/x=1

I tried many other methods, but I can't find any process.

I want to know rigorous proof of lim x→0 sin(x)/x=1 not using the differential and integral of trigonometrical function.

Thank you.

J.Doe
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  • Expand $\sin(x)$ as a series – asdf Apr 18 '18 at 10:51
  • Before you can prove anything at all about $\sin x$, you need a definition of $\sin x$. Tell us your definition, and then maybe we can answer your question. Or (even better) maybe you can answer it yourself. – GEdgar Apr 18 '18 at 11:35

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