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lim(x->∞) [ (x^3+x^2+x+1)^(1/3) -x ]
分子分母同时乘以 [ (x^3+x^2+x+1)^(2/3) +x.(x^3+x^2+x+1)^(1/3) + x^2 ]
=lim(x->∞) [ (x^3+x^2+x+1) -x^3 ]
/ [ (x^3+x^2+x+1)^(2/3) +x.(x^3+x^2+x+1)^(1/3) + x^2 ]
=lim(x->∞) (x^2+x+1)/ [ (x^3+x^2+x+1)^(2/3) +x.(x^3+x^2+x+1)^(1/3) + x^2 ]
分子分母同时除以 x^2
=lim(x->∞) (1+1/x+1/x^2)
/ [ (1+1/x+1/x^2+1/x^3)^(2/3) +(1+1/x +1/x^2+1/x^3)^(1/3) + 1 ]
=1/(1+1+1)
=1/3
分子分母同时乘以 [ (x^3+x^2+x+1)^(2/3) +x.(x^3+x^2+x+1)^(1/3) + x^2 ]
=lim(x->∞) [ (x^3+x^2+x+1) -x^3 ]
/ [ (x^3+x^2+x+1)^(2/3) +x.(x^3+x^2+x+1)^(1/3) + x^2 ]
=lim(x->∞) (x^2+x+1)/ [ (x^3+x^2+x+1)^(2/3) +x.(x^3+x^2+x+1)^(1/3) + x^2 ]
分子分母同时除以 x^2
=lim(x->∞) (1+1/x+1/x^2)
/ [ (1+1/x+1/x^2+1/x^3)^(2/3) +(1+1/x +1/x^2+1/x^3)^(1/3) + 1 ]
=1/(1+1+1)
=1/3
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