limx^2y/（x^2+y^2）,x趋向于0,y趋向于0

Question

limx^2y/（x^2+y^2）,x趋向于0,y趋向于0

Answer 1

分子分母同时除以x²
原式=lim(x,y→0,0) y/[1+(y/x)²]
∵(y/x)²≥0
∴1+(y/x)²>0
即1+(y/x)²≠0
∴lim(x,y→0,0) 1+(y/x)²≠0
又∵lim(x,y→0,0) y=0
∴原式=lim(x,y→0,0) y/[1+(y/x)²]=0