Question

多元函数求极限两题1 (x+y)/x^2-xy+y^2 xy趋向正无穷2 (绝对值x+绝对值y)/x^2+y^2 xy趋

Answer 1

1.∵x²+y²≥2|xy|

∴0≤|(x+y)/(x²+y²-xy)|=|x+y|/|x²+y²-xy|≤|x+y|/(x²+y²-|xy|)≤|x+y|/|xy|≤1/|x|+1/|y|

且lim(x,y->+∞)(1/|x|+1/|y|)=0

故有lim(x,y->+∞)[(x+y)/(x²+y²-xy)]=0

2.∵0≤(|x|+|y|)/(x²+y²)=|x|/(x²+y²)+|y|/(x²+y²)≤|x|/x²+|y|/y²=1/|x|+1/|y|

且lim(x,y->+∞)(1/|x|+1/|y|)=0

∴lim(x,y->+∞)[(|x|+|y|)/(x²+y²)]=0