求(e^x-e^y)/sinxy在（0,0）的极限

Question

求(e^x-e^y)/sinxy在（0,0）的极限

Answer 1

∵当y=x时,lim(x->0,y->0)[(e^x-e^y)/sin(xy)]=lim(x->0)[(e^x-e^x)/sin(x²)]
=lim(x->0)[0/sin(x²)]
=0
当y=0时,lim(x->0,y->0)[(e^x-e^y)/sin(xy)]=lim(x->0)[(e^x-e^0)/sin(0)]
=lim(x->0)[(e^x-1)/0]
=∞
∴说明x和y沿着不同的路径趋近于零时,(e^x-e^y)/sin(xy)的极限值都不相同
故(e^x-e^y)/sin(xy)在（0,0）的极限不存在.