What does $\lim\limits_{(x,y)\rightarrow0}$ mean and how to show $ \lim\limits_{(x,y)\rightarrow0}\frac{x^3}{x^2+y^2}=0$? Consider $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ where
$$f(x,y):=\begin{cases}
      \frac{x^3}{x^2+y^2} & \textit{ if }  (x,y)\neq (0,0) \\
      0 & \textit{ if }  (x,y)= (0,0)
    \end{cases} $$
If one wants to show the continuity of $f$, I mainly want to show that 
$$ \lim\limits_{(x,y)\rightarrow0}\frac{x^3}{x^2+y^2}=0$$
But what does $\lim\limits_{(x,y)\rightarrow0}$ mean? Is it equal to $\lim\limits_{(x,y)\rightarrow0}=\lim\limits_{||(x,y)||\rightarrow0}$ or does it mean $\lim\limits_{x\rightarrow0}\lim\limits_{y\rightarrow0}$?
If so, how does one show that the above function tends to zero?

$\lim_\limits{(x,y)\to 0}$ likely means $\lim_\limits{(x,y)\to(0,0)}$, which means that $x$ and $y$ are both tending to $0$. One could use polar coordinates where $x=r\cos(\theta)$ and $y=r\sin(\theta)$ to obtain:
$$\lim_{(x,y)\to(0,0)}\frac{x^3}{x^2+y^2}=\lim_{r\to 0}\frac{r^3\cos^3(\theta)}{r^2}=\lim_{r\to 0} r\cos^3(\theta)$$
Then note that $|\cos^3(\theta)|\leq 1~~~\forall\theta\in\Bbb R$.
