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?
 A: We have that


*

*for $x=y=t \to 0$
$$\frac{x^{2}+y^{2}}{x+y}=t \to 0$$


*

*for $x=t \to 0$ and $y=t^2-t$
$$\frac{x^{2}+y^{2}}{x+y}=\frac{t^2+t^2-2t^3+t^4}{t^2}=2-2t+t^2 \to 2$$
therefore the limit doesn't exist.

Edit
To solve these kind of limits at first we need to guess whether the limit exists or not and polar coordinates are often useful for that, notably in that case we obtain
$$\frac{x^{2}+y^{2}}{x+y}=r\frac{1}{\cos\theta+\sin \theta}=rf(\theta)$$
and since $f(\theta)$ is not bounded the limit seems prone to do not exist in that case.
Then the strategy to prove that is to find at least two different paths for which we obtain different limits as we have done.
In that case it easy to find the path for which the limit is equal to $0$, that is $x=y$. 
For the other path, a good strategy which often works, is to select at first a path such that the denominator is equal to zero that is $x=-y$ but since $f(x,y)$ is not defined at those points we add to $y$ an extra smaller term that is $t^2$. In some cases the first guess doesn't work and we need to use a different extra term for $y$.
A: 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)?.
