Proof of The Real Version
A simple proof is as follows.
For the first term, we note that ε⁄π(x2 + ε2) is a nascent delta function, and therefore approaches a Dirac delta function in the limit. Therefore, the first term equals ∓iπ f(0).
For the second term, we note that the factor x2⁄(x2 + ε2) approaches 1 for |x| ≫ ε, approaches 0 for |x| ≪ ε, and is exactly symmetric about 0. Therefore, in the limit, it turns the integral into a Cauchy principal value integral.
Read more about this topic: Sokhotski–Plemelj Theorem
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