Find the limit$,$ if it exists.

$\underset{(x,y)(0,0)}{lim}\frac{xy}{\sqrt[\phantom{2}]{x^2{y}^2}}$

Step $1$

To examine $\underset{(x,y)(\text{:}0,0\text{:})}{lim}\frac{xy}{\sqrt[\phantom{2}]{x^2{y}^2}},$ first approach $(0,0)$ along the $x-$axis.

On this path, all points have $y=0,y=0$

Stop $2$

Since $y=0$ for all points on this path, then we have

$\underset{(x,n)(0,0)}{lim}\frac{xy}{\sqrt[\phantom{2}]{x^2{y}^2}}=\underset{x0}{lim}\frac{x*0}{\sqrt[\phantom{2}]{x^{2}0}}=0\mathrm{.}0.$