Understanding a Professor's Limitation Equation

[BeauGeste]

My professor wrote on the board,
$$\lim_{\eta \rightarrow 0^+} \frac{1}{x-i \eta} = P(\frac{1}{x}) + i \pi \delta(x)$$
where P stands for principle value. I understand how the imaginary part comes about but why do you need P for the real part. Plus I thought Principle Value is defined for integrals that have singularities in them. Did he make an error when he wrote this?
thanks

 

[HallsofIvy]

??Well, obviously 1/x does have a singularity at x= 0!

 

[BeauGeste]

So what does P(1/x) mean? I understand what
$$P\int_{-\inf}^{\inf}\frac{1}{x} dx$$
means. When I saw principle value defined, it was operating on an integral that has a singularity. What does it mean for it to operate on a function with a singularity?