Limit of an Integral of a $C^\infty$-Smooth Function with Compact Support 

Let $\varphi:\R\rightarrow\R$ be a $C^\infty$-smooth function with compact support. Prove that the following limit exists, and compute the limit.
$$\lim_{\varepsilon\rightarrow0+} \int_{-\infty}^{\infty}\frac{\varphi(x)}{x+i\varepsilon} \;\mathrm{d}x ,      i=\sqrt{-1}$$


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  • Solving and writing up the solution to this question took a few hours. Please consider setting the price at a more appropriate level depending on the question's difficulty.

  • Hello. Sorry about that. First time using the service, so I didn't know what a good price would be. I also didn't know it would be so difficult to solve - I thought it would mostly use basic definitions since the other questions on this exam use mostly definitions or key but simply-stated theorems like the extreme value theorem. I would be willing to up the price but I don't think there is a way to do it after the fact?

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