Answer
(a) Doesn't exist. (b) Not continuous.
Work Step by Step
\[
(a) f(x, y)=\frac{x^{4}-x+y-x^{3} y}{-y+x}
\]
We will take the line $x=y$, so the line $x=y$ has parametric equations $x=$$t, y=t$ corresponding to $t=0$
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-x+y-x^{3} y}{-y+x}=\lim _{t \rightarrow 0} \frac{t^{4}-t+t-t^{3} t}{t-t}=\lim _{t \rightarrow 0} \frac{0}{0}=\text { undefined }
\]
Thus, the limit does not exist.
(b) According to definition (13.2.3), because $f$ is not defined at (0,0), it isn't continuous.
