Existance of multivariable limit

Question

We are asked to find whether the limit $$ \lim_{(x,y)\to(0,0)}\frac{x^3y^3}{x^8+y^4} $$ exists. I tried to find through the epsilon-delta definition but wasn't able to apply it successfully. So I tried to come up with a convergent sequence $(x_n)$ such that it converges to $(0,0)$ but $f(x_n)$ does not converge to a single point, thus establishing non-existence of the limit. But I wasn't able to come up with any such sequence. When I checked on Wolfram-Alpha it showed the limit to be non existent. Can someone provide a proof of this or give example of a sequence $(x_n)$ converging to $(0,0)$ such that such that $f(x_n)$ does not converge ?

Answer 1

By AM-GM inequality we have that

$$\frac{|x^3y^3|}{x^8+\frac{y^4+y^4+y^4}{3}}\leq \frac{3^{\frac{3}{4}}|x^3y^3|}{4|x^2y^3|} \leq |x|$$

thus the limit exists by squeeze theorem.