I am having problems proving that the limit of a certain multivariable function is equal to 0. What I need to prove is the following:
$$\lim_{(x,y)\rightarrow(0,0) }xy^2e^{x^2/y^4}=0$$ 
for $x,y \in D=\{(x,y):0\leq y \leq 1, 0\leq x\leq y^2\}$.
I tried solving the problem using the 'sandwich'theorem and ended up with the solution below: 
$$0\leq \lim_{(x,y)\rightarrow(0,0) }xy^2e^{x^2/y^4}\leq\lim_{(x,y)\rightarrow(0,0)} y^4e^{x^2/y^4}\leq\lim_{(x,y)\rightarrow(0,0) }y^4e^{x^2}= 0^4e^{0^2}=0\cdot1=0$$
It would be highly appreciated if someone could verify my answer, and perhaps give me some useful tips. 
Thank you 
ps: sorry for my bad English; I speak French.

This inequality is false:
$$\lim_{(x,y)\rightarrow(0,0)} y^4e^{x^2/y^4}\leq\lim_{(x,y)\rightarrow(0,0) }y^4e^{x^2}.$$
However, you can say that $e^{x^2/y^4}\le e^1$ (by assumption $x/y^2\le 1$) and write
$$\lim_{(x,y)\rightarrow(0,0)} y^4e^{x^2/y^4}\leq e \lim_{(x,y)\rightarrow(0,0) }y^4 =0.$$
