lim x趋向于0 [ln(1+x)/x]^[1/(e^x-1)]

来源:学生作业帮助网 编辑:作业帮 时间:2024/10/08 05:39:32
lim x趋向于0 [ln(1+x)/x]^[1/(e^x-1)]
x)Ux鄉Ov(DijWhWEkUj$VΆ4%5NC!'ITX1Ne`g[f?b>h ]2jMZdX3*Ԕ -1хjP06!X^&kod_\g ]Nz

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