<s>[INST] Solving $\lim\limits_{(x,y)\to(0,0)}\;\frac{x^5 + \,y^5}{x^3+\,y^3}$ How do I solve the limit $$\lim_{(x,y)\to(0,0)}\;\frac{x^5+y^5}{x^3+y^3}\quad ?$$ I have tried using polar coordinates, but I don't think an answer would be valid because theta is not fixed. What else can I do?
 [/INST] Hint
$$ \frac{x^5+y^5}{x^3+y^3} =\frac{x^5+x^2y^3}{x^3+y^3}+ \frac{x^3y^2+y^5}{x^3+y^3} - \frac{x^2y^2(x+y)}{x^3+y^3}  $$
And
$$x^2-xy+y^2 \geq |xy| \,.$$
OK, to make it more clear. If you combine the two hints, you get:
$$\left|\frac{x^5+y^5}{x^3+y^3}  \right| \leq x^2+y^2+|xy| \,.$$
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