Question

求lim┬(x→∞)∛(x^3-x^2-x+1)-x的具体过程.

求lim┬(x→∞)∛(x^3-x^2-x+1)-x的具体过程。

Answer 1

lim(x→∞)∛(x^3-x^2-x+1)-x=lim(x→∞)[∛(1-1/x-1/x^2+1/x^3)-1]/(1/x) 0/0型
=lim(1/3*(1/x+1/x^2-1/x^3) ' /[∛(1-1/x-1/x^2+1/x^3)]^2 ) /(-1/x^2)
=lim(1/3*(-1/x^2-2/x^3+3/x^4) /([∛(1-1/x-1/x^2+1/x^3)]^2 ) /(-1/x^2)
=lim(1/3*(-1/x^2-2/x^3+3/x^4)) /(-1/x^2) *lim 1/([∛(1-1/x-1/x^2+1/x^3)]^2 )
=1/3*1=1/3
计算过程嫌形式复杂 可以使用A=(1/x+1/x^2-1/x^3) 代换
并且有 lim(x→∞) A'/(-x^2)=1 lim(x→∞) 1/([∛(1-A)]^2 ) =1