∵x≠0,
∴[√(1+x^2+x^4)-√(1+x^4)]/x
=√(1/x^2+1+x^2)-√(1/x^2+x^2)
=[√(1/x^2+1+x^2)-√(1/x^2+x^2)]*[√(1/x^2+1+x^2)+√(1/x^2+x^2)]/[√(1/x^2+1+x^2)+√(1/x^2+x^2)]
=1/[1/x^2+1+x^2)+√(1/x^2+x^2)],
∵x≠0,∴(x-1/x)^2≥0,
即x^2-2+1/x^2≥0,
x^2+1/x^2≥2,
當(dāng)且僅當(dāng)x=1/x,x^2=1時(shí)取=號(hào),
∴√(1/x^2+1+x^2)≥√3,
√(1/x^2+x^2)≥√2,
√(1/x^2+1+x^2)+√(1/x^2+x^2)≥√3+√2,
∴1/[√(1/x^2+1+x^2)+√(1/x^2+x^2)]
≤1/(√3+√2)=√3-√2.
最大值=√3-√2.