How to prove the continuity of the function of several at $(0,0)$

Question

How to verify the continuity of the following function at $(0,0)$ :

\begin{cases}f(x,y)=\cfrac{x^3}{x^2+y^2}& \text{if } (x,y)\neq (0,0)\\ f(0,0)=0\end{cases}

Answer 1

In polar coordinates : $$\frac{x^3}{x^2+y^2}=\frac{r^3\cos^3\theta}{r^2}=r\cos^3\theta$$ And when $r\rightarrow 0,\forall \theta, f(r,\theta)\rightarrow0$.

So $f$ is indeed continuous.

Answer 2

Now ..I find another solution...$$|x^2|<|x^2+y^2|$$ $$\implies \frac{1}{|x^2+y^2|}\leq\frac{1}{|x^2|}$$ $$\implies\frac{|x^3|}{|x^2+y^2|}\leq |x|$$ And hence the given function is continuous at (0,0).