求极限lim(xy)^2/(x^2+y^2)^2,(x,y)趋于(0,0)

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求极限lim(xy)^2/(x^2+y^2)^2,(x,y)趋于(0,0)求极限lim(xy)^2/(x^2+y^2)^2,(x,y)趋于(0,0)求极限lim(xy)^2/(x^2+y^2)^2,(

求极限lim(xy)^2/(x^2+y^2)^2,(x,y)趋于(0,0)
求极限lim(xy)^2/(x^2+y^2)^2,(x,y)趋于(0,0)

求极限lim(xy)^2/(x^2+y^2)^2,(x,y)趋于(0,0)
lim[x=y,x-->0](xy)^2/(x^2+y^2)^2
=lim[x=y,x-->0]x^4/(4x^4)
=1/4
lim[y=2x,x-->0](xy)^2/(x^2+y^2)^2
=lim[y=2x,x-->0]4x^4/(25x^4)
=4/25
二者不相等
所以极限不存在

limxy/(x^2+y^2) (分子分母共同除以xy)
=lim1/[1/(x^2)+1/(y^2)]
因为lim1/(x^2)→无穷大
lim1/(y^2)→无穷大
故lim1/[1/(x^2)+1/(y^2)]=0
有limxy/(x^2+y^2)=0