How would one prove the following property for the dirac $\delta$ function ?
$$\delta'(x) = \lim_{a\to 0} \frac{\delta(x+2a) - \delta(x-a)}{a}$$
where $\delta$ is the famous function : $$\delta(x) = \begin{cases} +\infty & x=0, \\ 0 & x \neq 0 \end{cases}$$
Let $u$ be a smooth function with compact support and let the sequence of $(T_a)_a$distribution with $$T_a = \frac{\delta(x+2a)-\delta(x-a)}{a}$$
Then,
$$ \left( T_a,u\right) = \left( \frac{\delta(x+2a)-\delta(x-a)}{a}, u\right) = \frac{u(x+2a)-u(x-a)}{a} \\= 2\frac{u(x+2a)-u(x)}{2a}+\frac{u(x-a)-u(x)}{-a}\to 3u'(x)= -3(\delta'(x), u)$$
That $T_a\to -3\delta'(x)$