The requisite estimates are shown below.

If $8\left(x^2\&plus;y^2\right)$ is to be less than $\mathrm{\&epsilon;}$, and if $\mathrm{\&delta;}\&equals;\sqrt{x^2\&plus;y^2}$ is the radius of a circular neighborhood about the origin, then $8 {\mathrm{\&delta;}}^2<\mathrm{\&epsilon;}\Rightarrow \mathrm{\&delta;}\&equals;\sqrt{\mathrm{\&epsilon;}\&sol;8}$.

Figure 3.2.19(a) compares $8\left(x^2\&plus;y^2\right)$ with $\left|f\left(x\&comma;y\right)\right|$, the first in green, the second, in red. The green surface  lies above the red surface, indicating that near the origin, $8\left(x^2\&plus;y^2\right)$ is greater than $\left|f\left(x\&comma;y\right)\right|$.