Inverse in distributional sense

Question

Suppose that $\{f_n\}_{n\in\mathbb{N}}\subset\mathcal{C}^\infty(\mathbb{R})$ and $xf_n\to 1$ as $n\to\infty$ in distributional sense (take for example $f_n(x)=\frac{x}{x^2+(1/n)^2}$). Prove that $f_n\to P.V.\frac{1}{x}$, where $P.V.\frac{1}{x}$ is the Cauchy Principal Value of $1/x$, defined by $$ <P.V.\frac{1}{x},\phi>=\lim_{\epsilon\to0}\int_{\mathbb{R}\setminus[-\epsilon,\epsilon]}\frac{\phi(x)}{x}dx. $$ Any help is appreciate.