Answer
0
Work Step by Step
We find:
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{1-\cos \left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}
\]
Given that if $(x, y) \rightarrow(0,0)$ then $z \rightarrow 0$, so we re-write the equation:
\[
\begin{array}{l}
=\lim _{z \rightarrow 0} \frac{1-\cos z}{z} * \frac{(1+\cos z)}{1+\cos z} \\
=\lim _{z \rightarrow 0} \frac{1-\cos ^{2} z}{z}(1+\cos z) \\
=\lim _{z \rightarrow 0} \frac{\sin z}{z} \frac{\sin z}{(1+\cos z)} \\
=1 * 0 \\
=0
\end{array}
\]
