n^2*(x^1/n-x^1/n+1)n趋近于正无穷,x大于0求极限

Question

n^2*(x^1/n-x^1/n+1) n趋近于正无穷,x大于0 求极限

Answer 1

用Taylor展式：e^x=1+x+x^2/2+o(x^2),当x趋于0时.
于是x^(1/n)=e^(lnx/n)=1+lnx/n+(lnx)^2/(2n^2)+o(1/n^2)；
x^(1/(n+1))=e^(lnx/(n+1))=1+lnx/(n+1)+(lnx)^2/(2(n+1)^2)+o(1/n^2)；
因此有原极限（n趋于无穷）
=lim n^2(lnx/(n(n+1))+(lnx)^2/(2n^2)-(lnx)^2/(2(n+1)^2)+o(1/n^2))
=lnx.