i.
We consider (a,b)=(0,0)
Now the four different paths through (0,0) are
y=xy=x2y=x3y=x4
ii
f(x,y)={(x−a)4(y−)2(x−a)2(y−b)0if (x,y)=(a,b)if (x,y)=(a,b)But(a,b)=(0,0)f(x,y)={x4y2x2y0if (x,y)=(0,0)if (x,y)=(0,0)Now,lim(x,y)→(0,0)f(x,y)=lim(x,y)→(0,0)x4y2x2yy=xlim(x)→(0)x4∗x2x2∗x=0y=x2lim(x)→(0)x4∗x4x2∗x2=0.5y=x3lim(x)→(0)x4∗x6x2∗x3=0y=x4lim(x)→(0)x4∗x8x2∗x4=0
iii.
We will conclude that along every path chosen the limit of the function exist.
iv.
The function is continuous at the origin along every path chosen except along y=x2 7
v.
f(x,y)=x4+y2x2yfn=∂x∂f=(x4+y2)2(x2+y2)∗2xy−x2y(4x3)fn(0,0)=Nonexistanefn=∂y∂f=(x4+y2)2(x4+y2)∗x2−x2y(2y)fn(0,0)=Nonexistane
vi.
The function is differentiable at every point except at (0,0)
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