EXAMPLE 3 If f(x,y)=7x2+y45xy2, does lim(x,y)(0,0)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. Ther y=mx, where m is the slope, and f(x,y)=f(x,mx)=7x2+(mx)45x(72)2 =7x2+m4x4=7+m4x2 So f(x,y) as (x,y)(0,0) along y=mx. Thus f has the same limiting value along every nonvertical line through the origin. But that does not show that the given limit is 0 , for if we now let (x,y)(0,0) along the parabola x=y2, we have f(x,y)=f(y2,y)=7(y2)2+y4y2=8y4 So f(x,y) as (x,y)(0,0) along x=y2