BG-MO 1976 Problem 6

It is given a plane with a coordinate system with a beginning at the point $O$ . $A(n)$ , when $n$ is a natural number is a count of the points with whole coordinates which distances to $O$ are less than or equal to $n$ .

(a) Find $$\ell=\lim_{n\to\infty}\frac{A(n)}{n^2}.$$ (b) For which $\beta$ $(1<\beta<2)$ does the following limit exist? $$\lim_{n\to\infty}\frac{A(n)-\pi n^2}{n^\beta}$$