Transforming a function in the dirac delta function

Question

I need help finding $k(\sigma)$ such that the family of functions $$ \delta_\sigma (x,y) = k(\sigma) e^{-\frac{1}{2}\frac{x^2 + y^2}{\sigma^2}} $$ defines the unit impulse $\delta(x,y)$ as $\sigma \rightarrow 0.$

Answer 1

Here's a hint: Dirac delta approximators (also known as bounded approximate identities) have integral $1$ by definition. Try switching to polar coordinates when doing the integrals since you have a function of $x^2+y^2$ ($=r^2$).