Verify that the following definition of the delta function is valid I want to verify that the following is a valid definition of the delta function: 
$$\delta(x)=\lim_{a \to 0} \frac{1}{\pi}\frac{a}{a^2+x^2}$$
This satisfies $$\begin{cases} 0 & \text{if } x \neq 0 \\ \infty & \text{if } x=0 \end{cases}$$
I think I also need to verify that $\int_\mathbb{R} \delta(x) \, dx=1$. How do I do this?

$F(a) =\frac 1{\pi}\int_{-\infty}^{\infty} \frac{a}{a^2+x^2} dx = \frac {1}{\pi}\arctan \frac xa |_{-\infty}^\infty = 1$
$\lim_\limits{a\to 0} F(a) = \int_{-\infty}^{\infty} \delta (x)\ dx = 1$
