求极限((1-根号下x^2y+1)/x^3y^2)sin(xy),当x,y趋于0时

题目

求极限((1-根号下x^2y+1)/x^3y^2)sin(xy),当x,y趋于0时

答案

lim((1-√(x^2y+1))/x^3y^2)sin(xy),有理化1-√(x^2y+1))：
=lim(-x^2y)/(1+√(x^2y+1))/x^3y^2)sin(xy)
=lim(-sin(xy))/(1+√(x^2y+1))/(xy)
=-1/2 (利用limsin(xy)/(xy)=1)