Answer
$$0$$
Work Step by Step
Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{\sqrt{x^{2}+y^{2}}}$$
Let $y=mx$; then
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{\sqrt{x^{2}+y^{2}}}&=\lim _{x \rightarrow0} \frac{x^{2}-m^{2}x^2}{\sqrt{x^{2}+m^{2}x^2}}\\
&= \lim _{x \rightarrow0} \frac{x^{2}(1-m^{2})}{x\sqrt{1+m^{2}}}\\
&=\lim _{x \rightarrow0} \frac{x(1-m^{2})}{\sqrt{1+m^{2}}}\\
&=0
\end{align*}
