设f(0)=0,f'(x)在x=0的领域内连续,又f'(x)≠0证明:lim(x趋向0)x^f(x)=1

问题描述:

设f(0)=0,f'(x)在x=0的领域内连续,又f'(x)≠0证明:lim(x趋向0)x^f(x)=1

f'(0)=lim[f(x)-f(0)]/xlim(x趋向0)x^f(x)=e^[lim(x趋向0)f(x)lnx]=e^[lim(x趋向0)lnx/(1/f(x))]=e^[lim(x趋向0)1/x/(-f'(x)/f^2(x))]=e^[-lim(x趋向0)f^2(x)/(xf'(x))]=e^[-1/f'(0)lim(x趋向0)f^2(x)/(x)]=e^[-1/f'...