I know that classically the limit $$\lim_{x\to 0^+}{\sin \frac1x}$$ does not exist but how by the means of non-standard analysis one can prove that $$\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac12 \text{Ci}\left(\frac{1}{2}\right)\approx0.568318$$


Similar question about a proof that

$$\lim_{x\to0} \cos \frac1x=\frac 12 \text{Si}\left(\frac{1}{2}\right)-\frac{\pi }{4}+\cos \left(\frac{1}{2}\right)\approx 0.338738$$

  • $\begingroup$ i have no idea what $Ci$ is? $\endgroup$ – abel Mar 28 '15 at 10:22
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    $\begingroup$ What "non-standard analysis" ? $\endgroup$ – Timbuc Mar 28 '15 at 10:23
  • $\begingroup$ @abel Cosine integral mathworld.wolfram.com/CosineIntegral.html $\endgroup$ – Anixx Mar 28 '15 at 10:24
  • $\begingroup$ Anixx, thanks. learned something new. $\endgroup$ – abel Mar 28 '15 at 10:26
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    $\begingroup$ If you are asking for a proof, does that imply that you know the statement to be true? Can you give any reference where the equality comes from? $\endgroup$ – Martin R Mar 28 '15 at 10:31

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