Why can't we solve $\lim\limits_{(x, y) \to (0, 0)} \frac{xy}{\sqrt{x^2 + y^2}}$ through dividing by xy in both numerator and denominator? When solving this limit, we use methods like $\epsilon$ - $\delta$ definition and polar coordinate convertion. And I wonder if we can solve it by dividing by $xy$ in both numerator and denominator like this 
$\lim\limits_{(x, y) \to (0, 0)} \frac{xy}{\sqrt{x^2 + y^2}} = \lim\limits_{(x, y) \to (0, 0)} \frac{1}{\sqrt{\frac{1}{y^2} + \frac{1}{x^2}}} = 0$ 
Thanks in advance!

No, not really. The reason is the usual one that you can't divide by zero, so you cannot perform that division when $xy=0$. But that is on either coordinate axes. In particular it includes lots of points outside of just $(0,0)$.
You can see that the first expression is only undefined when both $x$ and $y$ are equal to zero, i.e. only at the point $(0,0)$ whereas the expression that you have written is undefined when either $x$ or $y$ is equal to zero.
