Solve the limit using polar coordinates

Question

I am given the limit :

$$ \lim_{(x,y)\to (0,0)}\left[ \ x^2 + y^2 \ln(x^2+y^2)\right] $$

to solve using polar coordinates first I would convert the equation to polar coordinates : $$ \lim_{(x,y)\to (0,0)} \left[\sin^2(\theta) + \cos^2(\theta) \ln(\sin^2(\theta) + \cos^2(\theta)) \right]$$

can I apply substitution to get:

$$ \ln(1) = 0$$

Answer 1

Important facts:

1. $$x=r\cos\theta$$
2. $$y=r\sin\theta$$
3. $$\begin{align}x^2+y^2&=r^2\cos^2\theta+r^2\sin^2\theta\\&=r^2(\cos^2\theta+\sin^2\theta)\\&=r^2\end{align}$$

This means you can substitute $r^2$ in for $x^2+y^2$, $r\cos\theta$ for $x$, $r\sin\theta$ for $y$, and $\lim_{r\to 0}$ in for $\lim_{(x,y)\to(0,0)}$.