证明lim(x->x0)三次根号x=三次根号x0

|x^(1/3)-x0^(1/3)|=|x-x0|/|x^(2/3)+x^(1/3)x0^(1/3)+x0^(2/3)|=|x-x0|/|(x^(1/3)+x0^(1/3)/2)^2+3x0^(2/3)/4|(配方)
≤|x-x0|/(3x0^(2/3)/4)<ε
可解出|x-x0|<(3x0^(2/3)/4)*ε
因此使|x^(1/3)-x0^(1/3)|<ε
只需需要使|x-x0|<(3x0^(2/3)/4)*ε
对任意ε>0令δ=(3x0^(2/3)/4)*ε
对任意x∈(x0-δ,x0+δ),都有|x^(1/3)-x0^(1/3)|<ε
因此lim(x->x0)三次根号x=三次根号x0