Answer
Limit does not exist
Work Step by Step
$(a)$
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{x-y}{x^{2}+y^{2}}
\]
So along $y=0$
\[
\begin{array}{l}
=\lim _{x \rightarrow 0} \frac{x-0}{x^{2}+0^{2}} \\
=\lim _{x \rightarrow 0} \frac{1}{x}
=\text{undefined}
\end{array}
\]
We can see that the limit does not exist.
(b)
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{\cos x y}{x^{2}+y^{2}}
\]
So along $y=0$
\[
\begin{array}{l}
=\lim _{x \rightarrow 0} \frac{\cos 0}{x^{2}+0^{2}} \\
=\lim _{x \rightarrow 0} \frac{1}{x^{2}} \\
=\text { not defined }
\end{array}
\]
We can see that the limit does not exist.
