limx趋于无穷(ln(1+x)/x)^(1/x)的极限
2个回答

lim[x→∞] {[ln(1 + x)]/x}^(1/x)

= lim[x→∞] [(1/x)ln(1 + x)]^(1/x)

= lim[x→∞] {ln[(1 + x)^(1/x)]}^(1/x)

= {lnlim[x→∞] [(1 + x)^(1/x)]}^{lim[x→∞] 1/x}

= [ln(e)]^(0)

= 1^0

= 1

如果不是上面那个,就是下面这个

lim[x→∞] {ln[(1 + x)/x]}^(1/x)

= lim[x→∞] [ln(1 + 1/x)]^(1/x)

lim[x→∞] (1/x)^(1/x)、取自然对数

= lim[x→∞] ln[(1/x)^(1/x)]

= lim[x→∞] ln(1/x)/x、洛必达法则

= lim[x→∞] 1/(1/x) * (- 1/x²)

= lim[x→∞] (- 1/x)

= 0

= ln(1)、去掉自然对数

= 1

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