Question

求一个二元函数的极限lim((x^2)y+x^5)/(x^4+x^6+2(x^3)y+y^2))(x,y)->(0,0)

Answer 1

极限不存在

设y=kx^2

代入得到：lim((x^2)y+x^5)/(x^4+x^6+2(x^3)y+y^2))

(x,y)->(0,0)

=lim(x->0)(kx^4+x^5)/(x^4+x^6+2kx^5+k^2x^4)

=lim(x->0)(k+x)/(1+x^2+2kx+k^2)

=k/(1+k^2)

极限与路径有关,所以,极限不存在