lim （x,y）->(0,0) xy/[根号下(xy+1)]-1的值为
7个回答

x^2+(y^2)/2=1,

x^2+[(1/√2)y]^2=1,

设x=cosA,y=√2sinA,

因x＞0,y＞0,不妨设0＜A＜π/2,

x√(1+y^2)=cosA√[1+2(sinA)^2]

=√{(cosA)^2[1+2(sinA)^2]}

=√{[1-(sinA)^2][1+2(sinA)^2]}

=√[1+(sinA)^2-2(sinA)^4]

=√{1+(1/√2)[√2(sinA)^2]-[√2(sinA)^2]^2}

=√{-[√2(sinA)^2-1/(2√2)]^2+[1/(2√2)]^2+1}

=√{-[√2(sinA)^2-√2/4]^2+9/8}

=√{-2[(sinA)^2-1/4]^2+9/8}

=√{-2[(sinA+1/2)(sinA-1/2)]^2+9/8}

≤√(9/8)=(3/4)√2

当sinA=1/2,即x=y=√2/2时,原式取最大值(3/4)√2.

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