How to evaluate $\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{x+y}$ I am supposed to evaluate the following limit:
$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{x+y}$$ 
My solution:
$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{x+y}
=\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}}{x+y}
+\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{y^{2}}{x+y}\\
=\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}x\cdot\frac{x}{x+y}
+\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}y\cdot\frac{y}{x+y}\\
=0+0=0$$
Can someone please tell me if it is correct?

The answer is wrong, that limit doesn't exist.
$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}}{x+y}=\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{x^{2}+y^{2}+2xy-2xy}{x+y}=\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}(x+y)-2\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{xy}{x+y}.$$
And
$$\lim_{\left ( x,y \right )\rightarrow \left ( 0,0 \right )}\frac{xy}{x+y}$$
dosen't exist as you can see from Does $\lim \frac{xy}{x+y}$ exist at (0,0)?.
