Calculate double limit with polar coordinates

Question

I have to calculate a double limit using polar coordintes. This is the limit: \begin{cases} \frac{x|y|}{\sqrt{x^2+y^2},} & (x,y)\neq (0,0), \\ 0, & (x,y)=(0,0). \end{cases}

I tried to resolve myself but I can´t continune.

Thanks for the help!

Answer 1

I'm guessing you have to calculate the limit at $0$. It is likely to be zero since it is defined as $0$ at $0$.

Anyhow, rewriting in polar:

we take the limit $\lim_{r\to 0}\frac{r\cos\theta * r|\sin\theta|}{r} = \lim_{r \to 0}(r\cos\theta*|\sin\theta|) = 0$