I am trying to show that $x\text{PV}\left(\frac{1}{x}\right) = 1$ in the sense of distributions, that is $\langle x\text{PV}\left(\frac{1}{x}\right), \phi \rangle = \langle 1, \phi \rangle$ for all $\phi$, where PV denotes the Cauchy Principle Value, and $\phi$ is a test function. Is the following correct?
$\langle x\text{PV}\left(\frac{1}{x}\right), \phi \rangle = \lim\limits_{\epsilon \to 0} \displaystyle\int_{|x| > \epsilon} x \left(\frac{1}{x}\right) \phi \ dx $.
If this is not the correct approach, what is?
edit:
So apparently the result is an immediate consequence of the definitions:
$\langle x\text{PV}\left(\frac{1}{x}\right), \phi \rangle = \langle \text{PV}\left(\frac{1}{x}\right), x \phi \rangle = \lim\limits_{\epsilon \to 0} \displaystyle\int_{|x| > \epsilon} \left(\frac{1}{x}\right) (x\phi) \ dx = \displaystyle\int_{\mathbb{R}} \phi \ dx = \langle 1, \phi \rangle$.