Q: Find the limit if it exists $$ \lim \limits_{(x,y) \to (0,0)} \frac{x^3 + y^3}{x^2 + y^2} $$
What I have done so far:
- Setting y = 0 & x = 0 I find that along these axes it looks like the limit will approach $0$ therefore: $$ \lim \limits_{(x,y) \to (0,0)} \frac{x^3 + y^3}{x^2 + y^2} = 0$$
$$\mid \frac{x^3 + y^3}{x^2 + y^2} - 0 \mid < \epsilon \quad \quad \quad 0 < \sqrt{x^2 + y^2} < \delta $$
Now since $ x^2 + y^2 \geq 0 $ we cans simplify to:
$$ \mid x^3 + y^3 \mid \leq \sqrt{ (x^3 + y^3)^2 } = \quad .... $$
I do not know how to proceed; how do I modify the $\sqrt{ (x^3 + y^3)^2 } $ into a $\sqrt{x^2 + y^2} $ ?