[Math] $\epsilon$-$\delta$ proof that $f(x) = x^3 /(x^2+y^2)$, $(x,y) \ne (0,0)$, is continuous at $(0,0)$

Question

I need to prove that $f$ continuous at $(x, y)=(0,0)$ using a $\epsilon$-$\delta$ proof
$$
f(x, y) = \begin{cases}
\frac{x^3}{{x^2 + y^2}},&(x,y)\neq (0,0)
\\
0,&(x,y) = (0,0)
\end{cases}
$$

I'm not sure how to manipulate the function to get $\delta$

Answer 1

You can use polar coordinates to see the problem more clearly: $$f(r,\theta)=\frac{r^3 \cos^3(\theta)}{r^2} = r \cos^3(\theta)$$ Thus, for any $\epsilon>0$, choose $\delta=\epsilon$. If $r<\delta$: $$|f(r,\theta)-0| < |r \cos^3(\theta)-0|<|r|<\delta=\epsilon$$