evaluating the limit $ \lim_{(x,y) \rightarrow (0,0) } \frac{x^2y^2}{x^2 + y^2} $ I was trying to evaluate $$ \lim_{(x,y) \rightarrow (0,0) } \frac{x^2y^2}{x^2 + y^2} $$ but only partially managed to do so through implying $y=kx$ but didn't find any other appropriate functions for this.
If you are familiar with the well-known inequality: $x^2+y^2\geq 2|xy|$, then you can use it: $0\leq \dfrac{x^2y^2}{x^2+y^2} = \left|\dfrac{xy}{2}\cdot \dfrac{2xy}{x^2+y^2}\right|=\dfrac{|xy|}{2}\cdot \dfrac{2|xy|}{x^2+y^2}\leq \dfrac{|xy|}{2}$, and since $|xy| \to 0$ when$x, y \to 0$, the squeeze theorem says the limit is $0$.
We have that
$$0\le\frac{x^2y^2}{x^2 + y^2}\le\frac{x^2y^2+y^4}{x^2 + y^2}=\frac{y^2(x^2+y^2)}{x^2 + y^2}=y^2\to 0$$