函数f(x)满足f(0)=0,f'(0)>0,则lim(x→0+)x的f(x)次方=

发布网友发布时间：2024-09-29 05:13

共1个回答

热心网友时间：2024-09-30 10:55

lim(x→0+)x^f(x)
=e^[lim(x→0+)f(x)lnx]
=e^[lim(x→0+)lnx/(1/f(x))]
=e^[lim(x→0+)1/x/(-f'(x)/f²(x))] (洛必达法则)
=e^[-lim(x→0+)f²(x)/(xf'(x))]
=e^[-1/f'(0)lim(x→0+)f²(x)/(x)]
=e^[-1/f'(0)lim(x→0+)2f(x)·f'(x)/(1)](再次洛必达法则)
=e^(-f(0)·f'(0)/f'(0)) (f'(0)>0)
=e^0
=1

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函数f(x)满足f(0)=0,f'(0)&gt;0,则lim(x→0+)x的f(x)次方=

函数f(x)满足f(0)=0,f'(0)>0,则lim(x→0+)x的f(x)次方=