Problem with limit 2 variables $\lim_{(x,y) \to (0,0)} \frac{x^2 y^2}{x^2 + y^2}$

Question

I tried to solve the limit:

$\lim \frac{x^2 y^2}{x^2 + y^2}$ with $(x,y) \to (0,0)$

I tried it with the paths $x=0, y=0, y=x, y=x^2$ and everything went to $0$.

Now I'm suspicious that this limit really goes to $0$, but how I prove it?

Thank you and sorry my english.

Answer 1

$|x| \leq \|(x,y)\|_2$ and $|y| \leq \|(x,y)\|_2$, then $$ |x^2 y^2| \leq \|(x,y)\|_2^4 $$ The denominator is exactly $\|(x,y)\|_2$ the limit follows.