lim(1+xy)*（1/x）,X趋于0,Y趋于0,但我不懂过程,我觉得是答案是e同学跟我讲了过程,呵呵

Question

lim(1+xy)*（1/x）,X趋于0,Y趋于0,但我不懂过程,我觉得是答案是e
同学跟我讲了过程,呵呵

Answer 1

lim(x->0,y->0)[(1+xy)^(1/x)]=lim(x->0,y->0)[(1+xy)^(y/xy)]
=lim(x->0,y->0){[(1+xy)^(1/(xy))]^y}
={lim(x->0,y->0)[(1+xy)^(1/(xy))]}^[lim(x->0,y->0)(y)]
={lim(t->0)[(1+t)^(1/t)]}^[lim(y->0)(y)] (令t=xy)
=e^[lim(y->0)(y)] (应用重要极限lim(z->0)[(1+z)^(1/z)]=e)
=e^0
=1.