<s>[INST] What is the limit of $\frac {x^4 +y^4}{x^3 +y^3}$ as $(x,y) \to(0,0)$ What is the limit of $$\lim_{(x,y)\to(0,0)} \dfrac {x^4 +y^4}{x^3 +y^3}$$
if it exists?
I have tried to solve it by converting it to polar system  $(x,y)=(r\cos a,r\sin a)$ and another settings. However I could not find the limit and not to show that there is no limit.
thanks in advance.
 [/INST] The idea to prove that the limit doesn't exist is to choose a direction such that $x^3+y^3$ is convergent to $0$ faster than $x^4+y^4$ so let $y=-x+x^3$ hence we find
$$x^3+y^3=x^3+(-x+x^3)^3=3x^5+o(x^5)\ \text{and}\ x^4+y^4=x^4+(-x+x^3)^4=2x^4+o(x^4)$$
hence
$$\lim_{(x,y=-x+x^3)\to(0,0)} \dfrac {x^4 +y^4}{x^3 +y^3}=\lim_{x\to0}\dfrac {2x^4 +o(x^4)}{3x^5 +o(x^5)}=\infty $$
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