f(x,y)=3xy/x^2+y^2 lim(x→0)(lim(y→0)f(x,y))

来源：学生作业帮助网编辑：作业帮时间：2024/04/29 10:30:08

f(x,y)=3xy/x^2+y^2 lim(x→0)(lim(y→0)f(x,y))
f(x,y)=3xy/x^2+y^2 lim(x→0)(lim(y→0)f(x,y))

f(x,y)=3xy/x^2+y^2 lim(x→0)(lim(y→0)f(x,y))
=0.
因为是先后取极限,而不是同时取极限.
当对y取极限时,x暂时看作常数.则lim(y→0)f(x,y)=3x×0/x^2+0^2=0.
再对x取极限仍得0.
如果是同时对x和y取极限,则极限不存在：
令y=k·x,k为常数,则
lim(x→0,y→0)=3kx²/(1+k²)x²
=3k/(1+k²)
当k取不同的值时,这个结果也会不同.也就是说不会得到一个确定的值.所以在这种情况下极限不存在.

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