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

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