Prove that $\eta^\epsilon(x)$ converges to Dirac delta

Question

I would like to prove that $$\eta^\epsilon(x):=\begin{cases}\displaystyle\frac{\epsilon-|x|}{\epsilon^2}&\text{, if }|x|\le \epsilon\\0&\text{, if }|x|\ge \epsilon\end{cases}$$ converges to Dirac delta if $\epsilon\to0$. How can I approach this problem? I observed that: $$\int \eta^\epsilon(x) dx=1$$ But how can I proceed?

Answer 1

Let $\phi\in C^\infty_C$. Then we have

$$\begin{align} \lim_{\varepsilon\to0}\int_{-\infty}^\infty \phi(x) \eta^\varepsilon (x) \,dx&=\lim_{\varepsilon\to0}\int_{-\varepsilon}^\varepsilon \phi(x) \frac{\varepsilon-|x|}{\varepsilon^2}\,dx\\\\ &\overbrace{=}^{x\mapsto \varepsilon x}\lim_{\varepsilon\to0}\int_{-1}^1 \phi(\varepsilon x)(1-|x|)\,dx\\\\ &=\phi(0)\int_{-1}^1 (1-|x|)\,dx\\\\ &=\phi(0) \end{align}$$

Therefore, in distribution we can write

$$\lim_{\varepsilon\to 0}\eta^\varepsilon(x)\sim \delta (x)$$