EXAMPLE 3 If f(x, y) = - (x,) =av21 A does l im f(x, y) exist? SOLUTION Let's try to save some time by letting (x, y) → (0,0) along any nonvertical line through the origin. Then y = mx, where m is the slope, and f(x, y) = f(x, mx) = 7x( mx )? 8x2 + (mx) 7m2x3 - 7mx 8x2 + m4x4 8 + m4x2 as (x, y) → (0,0) along y = mx. So f(x, y) → Thus f has the same limiting value along every nonvertical line through the origin. But that does not show that the given limit is o, for if we now let (x, y) → (0,0) along the parabola x = y2, we have f(x, y) = f(y2, y) = 8(82)2 + y4 So f(x, y) → as (x, y) → (0,0) along x = y2. Since different paths lead to different limiting values, the given limit does not exist.