高数limx趋于0f(x)/x=0,f''(0)=4,求limx趋于0[1+f(x)/x]^1/x

Question

如题所述

Answer 1

显然f(0)=f'(0)=0
ln原式=lim(x→0)ln(1+f(x)/x)/x
=lim(x→0)1/(1+f(x)/x)*(xf'(x)-f(x)/x^2
=lim(x→0)f'(x)/x-f(x)/x^2
=lim(x→0)(f'(x)-f'(0))/(x-0)-f'(x)/(2x)
=f''(0)-1/2lim(x→0)f'(x)/x
=f''(0)-f''(0)/2
=2
所以原式=e^2追问

怎么显然了，还有答案给的2

追答

搞笑吗？怎么会是2？你说是1我都觉得可能是我算错了……
因为lim(x→0)f(x)/x=0啊，所以lim(x→0)f(x)=0，即f(0)=0，所以f'(0)=0（导数定义）