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. 

Here's another way
$$ \lim\limits_{(x,y)\to(0,0)} \frac{(x+y)^2}{x^2+y^2} $$
Using polar coordinates, we have
$$ \lim\limits_{r\to 0^+} \frac{\left(r\cos\phi+r\sin\phi\right)^2}{r^2\cos^2\phi + r^2\sin^2\phi} $$
$$ = \lim\limits_{r\to 0^+} \frac{r^2\left(\cos\phi+\sin\phi\right)^2}{r^2\left(\cos^2\phi + \sin^2\phi\right)} $$
$$ = \lim\limits_{r\to 0^+} \frac{\left(\cos\phi+\sin\phi\right)^2}{\cos^2\phi + \sin^2\phi} $$
$$ = \lim\limits_{r\to 0^+} \left(\cos\phi+\sin\phi\right)^2 $$
$$ = \sin(2\phi)+1 $$
This limit is clearly dependent on $\phi$. Therefore
$$ \lim\limits_{(x,y)\to(0,0)} \frac{(x+y)^2}{x^2+y^2}\ \mbox{does not exist} $$
