<s>[INST] Prove, using first principles, that $\lim\limits_{(x,y)\to(0,0)}{x.y^2=0}$ My try is:
$\left|x.y^2-0\right|<\left|\sqrt{x^2+y^2}.(x^2+y^2)\right|< \delta.\delta^2=\epsilon$
where $0<\sqrt{x^2+y^2}< \delta$ what did I miss here?
 [/INST] Writing $f(x,y) = x\cdot y^2$. Let $\varepsilon > 0$, setting $\delta  = \varepsilon^{1/3}$,  you have that $\sqrt{x^2 + y^2} \leq \delta \implies |f(x,y)- 0| \leq (\sqrt{x^2 + y^2})(\sqrt{x^2 + y^2})^2 \leq \varepsilon^{1/3} \cdot \varepsilon^{2/3} = \varepsilon$. 
So, you have proven by constructing it that $\exists \delta, \sqrt{x^2 + y^2} \leq \delta \implies |f(x,y)- 0| \leq \varepsilon$.
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