求二元函数极限：（x,y)趋近于(2,-1/2)时lim(2+xy)^(1/(y+xy^2))

求二元函数极限：（x,y)趋近于(2,-1/2)时lim(2+xy)^(1/(y+xy^2))
求二元函数极限：（x,y)趋近于(2,-1/2)时lim(2+xy)^(1/(y+xy^2))

求二元函数极限：（x,y)趋近于(2,-1/2)时lim(2+xy)^(1/(y+xy^2))
取对数,得ln(2+xy)/(y+xy^2).
(x,y)→(2,-1/2),所以xy→-1,所以ln(2+xy)是无穷小,等价于1+xy.
所以,lim ln(2+xy)/(y+xy^2)=lim (1+xy)/(y+xy^2)=lim 1/y=-2.
所以,原极限是e^(-2).

=lim(1+(1+xy))^(1/(1+xy)*((1+xy)/(y+xy^2))=limexp(1/y)=exp(-2)