lim x趋向于0 [ln(1+x)/x]^[1/(e^x-1)]

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lim x趋向于0 [ln(1+x)/x]^[1/(e^x-1)]
lim x趋向于0 [ln(1+x)/x]^[1/(e^x-1)]

lim x趋向于0 [ln(1+x)/x]^[1/(e^x-1)]
原式=lim e^( ln[ln(1+x)/x] / (e^x-1))
=lim e^( ln[ln(1+x)/x] / x)
洛必达=lim e^[ (x-(1+x)ln(1+x)) / x(1+x)ln(1+x)]
=lim e^[ (x-(1+x)ln(1+x)) / (1+x)x²]
洛必达=lim e^[ -ln(1+x) /(3x²+2x)]
=lim e^[ -x /(3x²+2x)]
=lim e^[ -1 /(3x+2)]
=e^-1/2

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