Determine whether or not the limit exists: $\lim_{(x,y)\to(0,0)}\frac{(x+y)^2}{x^2+y^2}$ 
Determine whether or not the limit $$\lim_{(x,y)\to(0,0)}\frac{(x+y)^2}{x^2+y^2}$$
  exists. If it does, then calculate its value.

My attempt:
$$\begin{align}\lim \frac{(x+y)^2}{x^2+y^2} &= \lim \frac{x^2+y^2}{x^2+y^2} + \lim \frac {2xy}{x^2+y^2} =\\&= 1 + \lim \frac 2{xy^{-1}+yx^{-1}} = 1+ 2\cdot\lim \frac 1{xy^{-1}+yx^{-1}}\end{align}$$
But $\lim_{x\to 0^+} x^{-1} = +\infty$ and $\lim_{x\to 0^-} x^{-1} = -\infty$
Likewise, $\lim_{y\to 0^+} y^{-1} = +\infty$ and $\lim_{y\to 0^-} y^{-1} = -\infty$
So the left hand and right hand limits cannot be equal, and therefore the limit does not exist. 

If we approach $y=mx $ then $\lim_{(x,y)\to (0,0)}\frac{(x+y)^2}{x^2+y^2}
=\lim_{x\to 0}\frac{(x+xm)^2}{x^2+m^2x^2}=\lim_{x\to 0}\frac{(1+m)^2}{1+m^2}=\frac{(1+m)^2}{1+m^2} $ which is depends on $m$ so limit is not unique . So limit does not exist .
