fx(0,0)=h→0limhf(0+h,0)−f(0,0)
=h→0limh(0)2+h20⋅(h)3−0=0
fy(0,0)=h→0limhf(0,0+h)−f(0,0)
=h→0limhh2+(0)2h⋅(0)3−0=0
If (x,y)=(0,0)
fx(x,y)=(x2+y2)2y3(x2+y2−2x2)=(x2+y2)2y3(y2−x2)
Let x=rcosθ,y=rsinθ
(x,y)→(0,0)limfx(x,y)=(x,y)→(0,0)lim(x2+y2)2y3(y2−x2)
=r→0limr4r3sin3θ(r2sin2θ−r2cos2θ)
=r→0lim(rcosθ)2+(rsinθ)2rcosθ(rsinθ)3
=r→0limr2cosθsin3θ=0=f(0,0)
=r→0limrsin3θ(sin2θ−cos2θ)=0=fx(0,0)
The function fx(x,y) is continuous at (0,0).
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