Is there limit $ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$?

How to show if the limit $$ \lim_{(x,y) \to (0,0)} \frac{x^3}{x^2 + y^2}$$ exists? I suspect that there is, as I can't find any path that would show that limit doesn't exist, and WolframAlpha also suggests that the limit is (0,0).

In general, can you recommend any tips how to learn to approach similar limit problems (fractions and polynomials like this)? edit That is, excluding the polar coordinate conversion method?

$$ \left|\frac{x^3}{x^2+y^2}\right|\le|x| $$

We have to show that $\lim_{(x,y)\to (0,0)}f(x,y)=0$, so we have to prove the following claim:

$$\forall \epsilon>0,~~\exists\delta>0,~\forall(x,y)\left(0<||(x,y)-(0,0)||<\delta\Longrightarrow\left|\frac{x^3}{x^2+y^2}-0\right|<\epsilon\right)$$ We have $$||(x,y)-(0,0)||<\delta\Longrightarrow\sqrt{x^2+y^2}<\delta\longrightarrow|x|<\delta,~|y|<\delta$$ So if we set $z=\text{max}(|x|,|y|)$, then $z<\delta$ and from this we get: $$\left|\frac{x^3}{x^2+y^2}-0\right|=\frac{|x|^3}{|x|^2+|y|^2}<\frac{z^3}{z^2}=z<\delta$$ Therefore, it is sufficient that $\delta=\epsilon$.

Use polar coordinates to write the limit as

$$ \lim_{r \to 0} r \cos^3\theta $$