Evaluate $\displaystyle\lim_{(x,y)\to(0,0)}\dfrac{(x+y)^2}{x^2+y^2}$
Using polar, we have $x=r\cos(\theta),y=r\sin(\theta)$
Our limit becomes:
$$\lim_{r\to 0}\dfrac{(r\cos(\theta)+r\sin(\theta))^2}{r^2\sin^2(\theta)+r^2\cos(\theta)}=\lim_{r\to 0}\dfrac{r^2\cos^2(\theta)+2r^2\cos(\theta)\sin(\theta)+r^2\sin^2(\theta)}{r^2}$$
Factoring and dividing removes the $r^2$ in the denominator, and we get $1$ as the limit.
However this is not right.
If we consider along the $x-axis$, our limit becomes $1$. If we consider along the line $y=x$, our limit becomes $1/2$, and are clearly not equal.
This means that the limit does not exist but my polar said it does and it equals $1$. Where did I mess up in my polar coordinates?