<s>[INST] Find multivariable limit $\frac{x^2y}{x^2+y^3}$ Find multivariable limit of: $$\lim_{ \left( x,y\right) \rightarrow \left(0,0 \right)}\frac{x^2y}{x^2+y^3}$$
How to find that limit? I was trying to do the following, but i am not able to find a proper inequality:
$$| \frac{x^2y}{x^2+y^3} | = |y-\frac{y^4}{x^2+y^3}| \le$$ 
 [/INST] Let $a>b>0$, $x=a\cdot t$ and $b\cdot t$. Then we have 
$$
\lim_{t^+\to 0}\frac{a^2t^2\cdot bt}{a^2t^2 + b^3t^3}
=
\lim_{t^+\to 0}\frac{a^2bt^3}{t^3(a^2\frac{1}{t} + b^3)}
=
\lim_{t^+\to 0}\frac{a^2b}{(a^2\frac{1}{t} + b^3)}=+\infty
$$
and
$$
\lim_{t^-\to 0}\frac{a^2t^2\cdot bt}{a^2t^2 + b^3t^3}
=
\lim_{t^-\to 0}\frac{a^2bt^3}{t^3(a^2\frac{1}{t} + b^3)}
=
\lim_{t^-\to 0}\frac{a^2b}{(a^2\frac{1}{t} + b^3)}=-\infty
$$
</s>