$\underset{(x,y) \rightarrow (0,0)}{\text{lim}} \frac{xy}{y-x^3}$

Question

Evaluate

$$\underset{(x,y) \rightarrow (0,0)}{\text{lim}} \frac{xy}{y-x^3}$$

My attempt: I've tried to use polar coordinates $x=r\cos \theta, \; y = r \sin \theta$: $$\underset{(x,y) \rightarrow (0,0)}{\text{lim}} \frac{xy}{y-x^3} =\underset{r \rightarrow 0}{\text{lim}} \frac{r^2 \sin \theta \cos \theta}{r(\sin \theta - r^2 \cos \theta)} = 0 $$

but the book says it doesn't exists. What am I doing wrong?

Answer 1

Hint: See what happens if $(x,y)$ approaches $(0,0)$ along the curve $y=x^3+x^4$.

Remark: As to what you are doing wrong, implicitly you are assuming that $\theta$ is constant as $(x,y)$ approaches $(0,0)$, that we are approaching $(0,0)$ along a ray.