Two variable limit question

Question

The equation is:

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

So I used sandwich with $\frac{x}{x^2+y}$

And left with $x^2/y$ and I don't understand why its $0$.

Because I think if I take the paths $(x,y)\to(0,x^2)$ and $(x,y)\to(0,2x^2)$.

Its opposing the fact the the limit is $0$.

Hope you could help me out.

Answer 1

If you learn trigonometry, you can use it here. Put $x = r\cos \theta, y = r\sin \theta\implies r^2 = x^2+y^2, x^3 = r^3\cos^3\theta\implies \dfrac{x^3}{x^2+y^2} = \dfrac{r^3\cos^3\theta}{r^2}=r\cos^3\theta \to 0$ because $(x,y) \to 0 \iff r \to 0$.