Given a function $$\lim_{(x, y)\to(\infty, \infty)} \frac{x+y+2x^2+2y^2}{x^3+y^3}$$ find the limit or prove that it does not exist. To check whether the limit exists, the direction towards the point should be checked, but how one proceeds if both variable tends to infinity?
Substituting for $x = r\cos\theta, y = r\sin\theta$ $$\lim_{r \to\infty}\frac{r\cos\theta + r\sin\theta+2r^2\cos^2\theta + 2r^2\sin^2\theta}{r^3\cos\theta + r^3\sin\theta} = 0.$$
It shows that limit is zero, I am not sure if it's correct way to approach the problem.
The limit doesn't exist indeed let as $t \to \infty$
then
$$\frac{x+y+2x^2+2y^2}{x^3+y^3}=\frac{t-t+\frac1t+2t^2+2\left(t^2-2+\frac1{t^2}\right)}{t^3+\left(-t^3+3t-3\frac1t+\frac1{t^3}\right)}=\frac{4t^2-4+\frac1t+2\frac1{t^2}}{3t+3\frac1t+\frac1{t^3}}\to \infty$$
but for $x=y$ it is equal to $0$.