Question

求解两道多元函数求极限的题目

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Answer 1

(1)令y=kx,
x^2y^2/(x^2y^2+(x-y)^2)=k^2x^4/(k^2x^4+(1-k)^2x^2)
当k=1时此极限为1
当k=0.1时 此极限为0。
故此极限与k值有关。极限不存在。
(2)夹逼原则
[(xy)/(x+y)²]^(xy)≦[(xy)/(x²+y²)]^(xy)≦(xy/2xy)^(xy)
左边=(x,y)→(+∞,+∞) lim[(xy)/(x+y)²]^(xy)=(x,y)→(+∞,+∞) lim[(xy)/(x²+2xy+y²)]^(xy)
=(x,y)→(+∞,+∞) lim[1/(x/y+2+y/x)]^(xy)≦(x,y)→(+∞,+∞) lim(1/4)^(xy)=0
右边=(xy/2xy)^(xy)=(1/2)^(xy)=0
∴lim(x,y)→(+∞,+∞) [(xy)/(x²+y²)]^(xy)=0
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