Examine the behavior of $\displaystyle f(x,y) = \frac{6x^{2.1}}{x^{2}+y^{2}}$ as $(x,y)$ approaches $(0,0)$.

(a) Changing to polar coordinates, we find

$\displaystyle \lim_{(x,y) \to (0,0)} \Big( \frac{6x^{2.1}}{x^{2}+y^{2}} \Big) = \lim_{r \to 0^+,\ \theta = anything} \Big($ $\Big) =$ .

Use "theta" for $\theta$. Use "infinity" for "$\infty$" and "-infinity" for "$-\infty$". Use "DNE" for "Does not exist".

(b) Since $f(0,0)$ is undefined, $f$ has a discontinuity at $(x,y) = (0,0)$. Is it possible to define a function $g : \mathbb{R}^2 \to \mathbb{R}$ such that $g(x,y) = f(x,y)$ for all $(x,y) \ne (0,0)$ and $g$ is continuous everywhere? If so, what would the value of $g(0,0)$ be? If there is no continuous function $g$, enter DNE.

$g(0,0) =$