Find $\lim\limits_{,→0} (^2+^2) \ln(^2+^2)$

Question

How do I find this limit using polar coordinates? I get to $r^2 \ln(r^2)$ but don't know what to do after that.

Answer 1

Hint

Prove$$\lim_{u\to 0^+}u\ln u=0$$

Here is a sketch:

Answer 2

To find $$\lim_{r \rightarrow 0}r^{2} \ln(r^{2}) = \lim_{r \rightarrow 0}2r^{2} \ln(r) = \lim_{r \rightarrow 0}2 \frac{\ln(r)}{r^{-2}}\text.$$ Applying L'hopital's rule, since $0 \cdot -\infty$ is an indeterminate form, we get $$\lim_{r \rightarrow 0}r^{2} \ln(r^{2}) =\lim_{r \rightarrow 0}2 \frac{\ln(r)}{r^{-2}} = \lim_{r \rightarrow 0}2 \frac{r^{-1}}{-2r^{-3}} =-\lim_{r \rightarrow 0} r^{2} = 0\text.$$