求lim n趋向于无穷 n(n^1/n -1)/ln n

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求lim n趋向于无穷 n(n^1/n -1)/ln n

求lim n趋向于无穷 n(n^1/n -1)/ln n
求lim n趋向于无穷 n(n^1/n -1)/ln n

求lim n趋向于无穷 n(n^1/n -1)/ln n
令t=1/n 则:t→0
lim n(n^1/n -1)/ln n
=lim﹣(1/t^t -1)/tlnt
=lim(1- 1/t^t )/lnt^t
∵t→0
∴可求 t→0时,t^t 极限为 1
令x=t^t,则x→1
∴原始化为:
lim(1- 1/x)/lnx (0/0型,用罗比达法则)
=lim(1- 1/x)'/(lnx)'
=lim(1/x²)/(1/x)
=lim1/x
=1
综上,原式极限为 1