求lim n趋向正无穷 n+(n^2-n^3)^1⼀3

解法一：原式=lim(n->∞){[n+(n²-n³)^(1/3)][n²-n(n²-n³)^(1/3)+(n²-n³)^(2/3)]/[n²-n(n²-n³)^(1/3)+(n²-n³)^(2/3)]}
=lim(n->∞){[n³+(n²-n³)]/[n²-n(n²-n³)^(1/3)+(n²-n³)^(2/3)]} (应用两数立方和公式)
=lim(n->∞){n²/[n²-n(n²-n³)^(1/3)+(n²-n³)^(2/3)]}
=lim(n->∞){1/[1-(1/n-1)^(1/3)+(1/n-1)^(2/3)]} (分子分母同除n²)
=1/[1-(0-1)^(1/3)+(0-1)^(2/3)]
=1/3；
解法二：原式=lim(n->∞){[1+(1/n-1)^(1/3)]/(1/n)} (分子分母同除n)
=lim(n->∞){[(1/3)(-1/n²)/(1/n-1)^(2/3)]/(-1/n²)} (0/0型极限，应用洛必达法则)
=lim(n->∞)[(1/3)/(1/n-1)^(2/3)]
=(1/3)/(0-1)^(2/3)
=1/3。

----------------------------分子分母同除以n
lim [n+(n^2-n^3)^(1/3)] = lim { 1+[(1/n)-1]^(1/3)}/(1/n)
n->+∞ n->+∞
洛必达法则
=lim [1+(x-1)^(1/3)]/x = lim(1/3)(x-1)^(-2/3)=1/3
x->0 x->0

=n+n*(1/n-1)^(1/3)
=(1+(1/n-1)^(1/3))/(1/n)
=(1+(x-1)^(1/3))/x .....(x趋向于0正)
=(1/3)*(x-1)^(-2/3）......(洛必达法则)
=1/3