Does the following limit exist?
$$\lim_{(x,y) \to (0,0)}\frac{x^2y^3}{x^4+(x^2+y^3)^2}$$
I tried to solve this problem using polar coordinates, but I can't simplify it. I tried the squeeze theorem, I got $0.5$, but I think this is incorrect.
[Math] Multivariable limit of rational function
limitsmultivariable-calculus
Best Answer
Let $x^2=y^3$ therefore we have $$\lim\limits_{(x,y)\to(0,0)}\frac{x^2y^3}{x^4+(x^2+y^3)^2}=\lim\limits_{x\to0}\frac{x^4}{5x^4}=\frac{1}{5} \ \ \ (1)$$ and if we consider $x=y$ then $$\lim\limits_{(x,y)\to(0,0)}\frac{x^2y^3}{x^4+(x^2+y^3)^2}=0 \ \ \ \ (2)$$ so that from (1) and (2) we see that limit doesn't exist.