Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$

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 $$\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac12 \text{Ci}\left(\frac{1}{2}\right)\approx0.568318$$

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Similar question about a proof that

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

• i have no idea what $Ci$ is? – abel Mar 28 '15 at 10:22
• What "non-standard analysis" ? – Timbuc Mar 28 '15 at 10:23
• @abel Cosine integral mathworld.wolfram.com/CosineIntegral.html – Anixx Mar 28 '15 at 10:24
• Anixx, thanks. learned something new. – abel Mar 28 '15 at 10:26
• 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? – Martin R Mar 28 '15 at 10:31