Two variables function limit In an exercise of a limit of a function of two variables in the solution I read this inequality:
$$ \frac{x^2y^2}{(x^2+y^2)^\frac{3}{2}} \le \frac{1}{2}\sqrt{x^2+y^2} $$
how did they arrive at this result?

That inequality is equivalent to$$x^2y^2\leqslant\frac12\sqrt{x^2+y^2}(x^2+y^2)^{3/2}=\frac{(x^2+y^2)^2}2.$$Besides,$$(x^2+y^2)^2-2x^2y^2=x^4+y^4\geqslant0.$$
