设f(x)在0到正无穷大上可导,f(x)>0,limf(x)=1(x趋向正无穷大),若lim[f(x+nx)/f(x)]^(1/n)(n趋向于0)

问题描述:

设f(x)在0到正无穷大上可导,f(x)>0,limf(x)=1(x趋向正无穷大),若lim[f(x+nx)/f(x)]^(1/n)(n趋向于0)
=e^(1/x),求f(x)

证:由lim[f(x+nx)/f(x)]^(1/n)=e^(1/x),(n趋向于0)
得e^[f(x+nx)-f(x)]/f(x)*(1/n)=e^(1/x),),(n趋向于0)
得lim[f(x+nx)-f(x)]/nf(x)=1/x 用罗比达法则:
limx*f'(x+nx)/f(x)=1/x(n趋向于0)又f(x)>0
得f'(x)/f(x)=1/x^2
f(x)=e^-(1/x)+c limf(x)=1(x趋向正无穷大)求得C=0
故f(x)=e^-(1/x)