求极限lim(x→+∞)x^1/3[(x+1)^2/3-(x-1)^2/3]请不要用洛必达法则

Answer 1

解：（泰勒公式法）
∵(1+1/x)^(2/3)=1+2/(3x)-1/(9x²)+o(1/x²) (符号o(x)表示高阶无穷小)
(1-1/x)^(2/3)=1-2/(3x)-1/(9x²)+o(1/x²)
∴(1+1/x)^(2/3)-(1-1/x)^(2/3)=4/(3x)+o(1/x²)
故原式=lim(x->+∞){x^(1/3)[(x(1+1/x))^(2/3)-(x(1-1/x))^(2/3)]}
=lim(x->+∞){x[(1+1/x)^(2/3)-(1-1/x)^(2/3)]}
=lim(x->+∞){x[4/(3x)+o(1/x²)]}
=lim(x->+∞)[4/3+o(1/x)]
=lim(x->+∞)(4/3)+lim(x->+∞)[o(1/x)]
=4/3+0
=4/3