Problem with limits when using polar coordinates:

Question

If I did the following limit using x = rcos y = rsin I find that the limit goes to 0. Doesn't that prove that the limit does not exist since I found 2 different limits ?

Answer 1

$$\lim _{ r\rightarrow 0 }{ \frac { 3{ r }^{ 2 } }{ \sqrt { { r }^{ 2 }+4 } -2 } } =\lim _{ r\rightarrow 0 }{ \frac { 3{ r }^{ 2 }\left( \sqrt { { r }^{ 2 }+4 } +2 \right) }{ { r }^{ 2 } } } =12$$

Answer 2

When you make the transformation, you should get

$\displaystyle \lim_{r \to 0} \frac{3r^2}{\sqrt{r^2+4}-2} $.

Now conjugate the denominator to get $\displaystyle \lim_{r \to 0} \frac{3r^2\left(\sqrt{r^2+4}+2\right)}{r^2}=\displaystyle \lim_{r \to 0} 3(\sqrt{r^2+4}+2) =\boxed{12}$