Answer
$$2$$
Work Step by Step
Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}$$
Rewriting, we have
\begin{align*}f(x,y)&=\frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}\\
&=\frac{\left(x^{2}+y^{2}\right)(\sqrt{x^{2}+y^{2}+1}+1)}{(\sqrt{x^{2}+y^{2}+1}-1)(\sqrt{x^{2}+y^{2}+1}+1)}\\
&=\frac{\left(x^{2}+y^{2}\right)(\sqrt{x^{2}+y^{2}+1}+1)}{\left(x^{2}+y^{2}+1-1\right)}\\
&=\sqrt{x^{2}+y^{2}+1}+1
\end{align*}
Then
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}+y^{2}}{\sqrt{x^{2}+y^{2}+1}-1}&=\lim _{(x, y) \rightarrow(0,0)} \sqrt{x^{2}+y^{2}+1}+1\\
&=2
\end{align*}
