I'm trying to solve the following excercise, whitout any luck.
Let $\eta(x)=\begin{cases} c \exp\left(\dfrac{1}{|x|^2-1}\right), & \text{if} \;|x| \leq 1 \\ 0 & \end{cases} $ $\qquad \text{for }x\in \mathbb R^d$
$c$ is chosen such that $\int_{R^d}\eta~dx=1$
Set now $\eta_\epsilon(x)=\dfrac{1}{\epsilon^d}\eta\left(\dfrac{|x|}{\epsilon}\right)$ (I think it looks strange but there is an actual absoulte value on $x$)
Show that $$ \lim\limits_{\epsilon \to 0} \eta_\epsilon=\delta \qquad \text{in } \mathcal D'(R^d) $$
We can write by definition for a test function $\phi$
$$\langle \eta_\epsilon,\phi\rangle = \int_{\Bbb R^d} \frac{c}{\epsilon^d} \exp\left(\dfrac{1}{|x|^2/\epsilon^2-1}\right)\phi(x)dx = \int_{\Bbb R^d} {c} \exp\left(\dfrac{1}{|y|^2 -1}\right)\phi(y\epsilon)dy.$$ By dominated convergence theorem the latter intergral converges to $\phi(0)$ as $\epsilon\to 0$, hence $\eta_\epsilon\to\delta_0$.