<s>[INST] Calculate the limit of a function of two variables How to calculate this limit?
$$\lim_{(x,y) \rightarrow (0,0)}\frac{x^2+y^2}{\sqrt{x^2+y^2+9}-3}$$
 [/INST] You have for $(x,y) \neq (0,0)$:
\begin{align}
\frac{x^2+y^2}{\sqrt{x^2+y^2+9}-3}
=\frac{(x^2+y^2)}{\sqrt{x^2+y^2+9}-3} \cdot \frac{\sqrt{x^2+y^2+9}+3}{\sqrt{x^2+y^2+9}+3}
=\sqrt{x^2+y^2+9}+3
\end{align}
Hence:
\begin{align}
\lim_{(x,y) \rightarrow (0,0)}\frac{x^2+y^2}{\sqrt{x^2+y^2+9}-3}
=\lim_{(x,y) \rightarrow (0,0)}\sqrt{x^2+y^2+9}+3=\color{blue}{6}
\end{align}
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