Evaluate limit (x,y) -- (0,0) [x^2+xy+y^2]/[x^2+y^2] by switching from Cartesian coordinates to polar

Question

Evaluate limit (x,y) --> (0,0)  [x^2+xy+y^2]/[x^2+y^2] by switching from Cartesian coordinates to polar

anonymous · Answer

$$\lim_{(x,y)\to(0,0)}\frac{x^2+xy+y^2}{x^2+y^2}=\lim_{(r,\theta)\to(0,0)}\frac{r^2+r^2(\sin\theta+\cos\theta)}{r^2}\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\lim_{(r,\theta)\to(0,0)}(1+\sin\theta\cos\theta)\
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\lim_{\theta\to0}(1+\sin\theta\cos\theta)
$$

anonymous · Answer

so does  the limit exist?

anonymous · Answer

Hmm, not according to WolframAlpha... http://www.wolframalpha.com/input/?i=Limit+%28x^2%2Bxy%2By^2%29%2F%28x^2%2By^2%29+as+%28x%2Cy%29+approaches+%280%2C0%29 Sorry, I'm not sure what's wrong here. I think it might have something to do with path dependence. There must be some function of $\theta$ that depends on $r$ for which the conversion doesn't directly apply.