Use the Two Path Test to prove that the following limit does not exist.

$\underset{(xy)(0,0)}{lim}\frac{6xy}{4x^2{y}^2}$

Suppose $(x,y)$ approaches $(0,0)$ along the line $y=mx$ for a constant $m.$ Evaluate the limit along this line for arbitrary $m.$ Substitute mx for $y$ and evaluate.

$\underset{(xy)(0,0)}{lim}\frac{6xy}{4x^2{y}^2}=6m$

As $(x,y)$ approaches $(0,0)$ along the path where $m=1,$ fx$,$ y$)$ approaches $,$ and as $(x,y)$ approaches $(0,0)$ along the path where $m=-1,$ fix.y$)$ approaches

$\backslash$table$[[\frac{6}{5}$