Question

请教一道求极限的题目

求(x,y)趋近于(0,0)时，lim((x^3)+(x^2)y+x(y^2)+(y^3))/((x^2)-xy+(y^2))

Answer 1

令：x=rcosθ ， y=rsinθ

lim[(x,y)->(0,0)] ((x^3)+(x^2)y+x(y^2)+(y^3))/((x^2)-xy+(y^2))
=lim[(x,y)->(0,0)] (r^3*(sinθ+cosθ)/(r^2-r^2sinθcosθ)
=lim[(x,y)->(0,0)] r* {(sinθ+cosθ)/(1-sinθcosθ)}
=lim[(x,y)->(0,0)] r* {(sinθ+cosθ)/(1-1/2*sin2θ)}
∵ 1/2 ≤ |1-1/2*sin2θ| ; |{(sinθ+cosθ)/(1-1/2*sin2θ)}|≤2/(1/2)=4
=0