求极限lim(x,y)→(+∞,+∞) [(xy)/(x^2+y^2)]^xy.
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求极限lim(x,y)→(+∞,+∞) [(xy)/(x²+y²)]^(xy)
解：[(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
解：[(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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