# dirac delta function limit form equality

Show that $$\lim_{y\to\infty}\frac{1}{\pi}\frac{y}{y^2+x^2} = \delta(x)$$

I do not know where the $\pi$ arise.

• I don't know either shouldn't that limit always equal to 0? – gamma Oct 30 '15 at 2:58
• @frank000: The limit is not to be taken pointwise, but in the usual topology on the space of distributions. – Nate Eldredge Oct 30 '15 at 3:03
• Hint: what do you get if you integrate the function on the right side over $\mathbb{R}$ (essentially pairing it, as a distribution, with the function 1)? What do you get if you leave out the $\pi$? – Nate Eldredge Oct 30 '15 at 3:05

The constant $\pi$ is to normalize the area under the curve to unity.
$$\int_{-\infty}^{\infty}{\frac{y}{y^2+x^2}}dx = \pi$$