Finding limit in $\mathcal{D}'(\mathbb{R})$

Question

I tried to solve the following limit in $\mathcal{D}'(\mathbb{R})$: \begin{equation} \lim_{\epsilon \to 0} \frac{\epsilon}{2} \left|x \right|^{\epsilon -1} \end{equation}

I began by thinking of testing on a $\phi \in \mathcal{D} (\mathbb{R})$, but all my approaches (part integration, thinking of a derivative anywhere) were unsuccessful...any help is welcome, thank you in advance!

Answer 1

$$f_\epsilon(x)=\frac12 sign(x) |x|^\epsilon \to \frac12 sign(x)$$

$$f_\epsilon' \to \delta$$