Use the Two Path Test to prove that the following limit does not exist.
\lim_((xy)->(0,0))(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.
\lim_((xy)->(0,0))(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
\table[[(6)/(5)