∵ lim x^(1/x)
x→∞
=lim e^[lnx^(1/x)]
x→∞
=lim e^[(1/x)lnx]
x→∞
=lim e^[1/x] = 1
x→∞
∴ lim (1+x)^(1/x)
x→∞
∴ lim (1+x)^{[1/(x+1)][(x+1)/x]}
x→∞
=1^1 = 1

∴ lim [(1+x)^(1/x) - e]/x = (1 - e)/ ∞ = 0
x→∞
樓主已經(jīng)更正,按照新的極限要求,重新解答如下：
∵ lim (1+x)^(1/x) = e
x→0
∴ lim [(1+x)^(1/x) - e]/x (0/0型)
x→0
=lim {[(1+x)^(1/x)][x/(1+x) - ln(1+x)]/x² - 0}/1
x→0
=lim e[x/(1+x) - ln(1+x)]/x²
x→0
=lim e[x - (1+x)ln(1+x)]/(x²+ x³)(0/0型)
x→0
=lim e[1 - ln(1+x) - 1]/(2x + 3x²)
x→0
=lim -eln(1+x)/(2x + 3x²) (0/0型)
x→0
=lim -e/[(1+x)(2 + 6x)]
x→0
= -e/2