Let

$$g_{\epsilon}(x)=\frac{-2 \epsilon x}{\pi\left(\epsilon^{2}+x^{2}\right)^{2}} .$$

By considering the integral $\int_{-\infty}^{\infty} \phi(x) g_{\epsilon}(x) d x$, where $\phi$ is a smooth, bounded function that vanishes sufficiently rapidly as $|x| \rightarrow \infty$, identify $\lim _{\epsilon \rightarrow 0} g_{\epsilon}(x)$ in terms of a generalized function.