I have read that
$$ \lim_{\substack{t \to \infty \\ \eta \to 0 \\ \eta t \to 0}} \frac{e^{-i x_0 t}}{x_0 - x +i\eta} = -2\pi i \delta (x - x_0) $$
where $\delta$ is Dirac delta and the limit is meant in this sense. Is this true? How do I prove it? How do I get the prefactor $-2 \pi i$?