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.
$$\lim_{\varepsilon\rightarrow0+} \int_{\infty}^{\infty}\frac{\varphi(x)}{x+i\varepsilon} \;\mathrm{d}x , i=\sqrt{1}$$
Answer
 The questioner was satisfied and accepted the answer, or
 The answer was disputed, but the judge evaluated it as 100% correct.
1 Attachment

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 simplystated 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?
 answered
 114 views
 $10.00
Related Questions
 Mathematical modeling
 real analysis
 Calculating P values from data.
 Prove the uniqueness of a sequence using a norm inequality.
 Need Upper Bound of an Integral
 How do I compare categorical data with multiple uneven populations?
 Prove Holdercontinuity for $\mu_\lambda (x) = \sum\limits_{n=1}^\infty \frac{ \cos(2^n x)}{2^{n \lambda} }$
 Let $(X, \cdot)$ be a normed space. Let $\{x_n\}$ and $\{y_n\}$ be two Cauchy sequences in X. Show that the seqience Show that the sequence $λ_n = x_n − y_n $ converges.