This is Part 6 (last part) of a problem statement of an old comprehensive exam question that I am working on.
It asks to evaluate
$$\lim_{r_0 \to 0} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{f(x,y)k(r_0)^2}{\pi(x^2+y^2+r_0^2)}dxdy$$
where $f(x,y)$ is a bounded and continuous function of $x$ and $y$.
From a previous part of this problem, I computed the limit of $\large \frac{k(r_0)^2}{\pi(x^2+y^2+r_0^2)}$, which is equal to $0$ for all $(x,y)\ne(0,0)$, and equal to $\infty$ at $(x,y) = (0,0)$.
Also, I computed the integral of $\large \frac{k(r_0)^2}{\pi(x^2+y^2+r_0^2)}$ inside the disk of radius $R$, centered at $(0,0)$, and got $\large \frac{kR^2}{R^2-r_0^2}$.
So, now we see that the integral of $\large \frac{k(r_0)^2}{\pi(x^2+y^2+r_0^2)}$ $\to$ $k$, as $r_0 \to 0$. I.e., we get a bump of $k$ in a neighborhood of $(0,0)$.
Now, back to the above limit that I want to evaluate. A simple check of the integrand shows that we can apply the Dominated Convergence Theorem for $(x,y)\ne(0,0)$. Then we have that
$$\lim_{r_0 \to 0} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\frac{f(x,y)k(r_0)^2}{\pi(x^2+y^2+r_0^2)}dxdy = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\lim_{r_0 \to 0}\frac{f(x,y)k(r_0)^2}{\pi(x^2+y^2+r_0^2)}dxdy$$
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty}0dxdy = 0 $$
for all $(x,y) \ne (0,0)$. The limit then reduces to just evaluating
$$\lim_{r_0 \to 0} \int \int_{B_\delta (0,0)} \frac{f(x,y)k(r_0)^2}{\pi(x^2+y^2+r_0^2)}dxdy$$
But I am stuck here and don't know how to show it. I think I need to somehow use the continuity and boundedness of $f$.
A student solution claims the answer is $f(0,0)k$, but I do not know how to arrive at this number yet.
Any ideas are welcome.
Thanks,