Answer
(a) $\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0$
(b) f is continuous at (0,0).
Work Step by Step
$$
(a) f(x, y)=\left\{\begin{array}{cl}\frac{x^{4}-y^{4}}{x^{2}-y^{2}} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{array}\right.
$$
$$
\lim _{(x, y) \rightarrow(0,0)} f(x, y)=\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{2}-y^{2}}=\lim _{(x, y) \rightarrow(0,0)} \frac{\left(x^{2}-y^{2}\right)\left(x^{2}+y^{2}\right)}{x^{2}-y^{2}}
$$
The function approaches 0 when $(x, y)$ approaches (0,0).
\[
0=\lim _{(x, y) \rightarrow(0,0)} f(x, y)
\]
(b) We found that:
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{2}-y^{2}}=\lim _{(x, y) \rightarrow(0,0)} 0=y^{2}+x^{2}
\]
is defined at (0,0) and $\lim _{(x, y) \rightarrow(0,0)} f(x, y)=0=f(0,0)$
F is continuous at (0,0) according to definition $13.2 .3 .$
