Question

请帮忙：limx趋于无穷(ln(1+x)/x)^(1/x)的极限

Answer 1

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

Answer 2

.