设f(x)具有连续的二阶可导,且f(0)二阶导=4,lim（x->0)f(x)/x=0,则lim(x->0)(1+f(x)/x)^1/x=?

Question

设f(x)具有连续的二阶可导,且f(0)二阶导=4,lim（x->0)f(x)/x=0,则lim(x->0)(1+f(x)/x)^1/x=?

Answer 1

lim（x->0)f(x)/x=0,所以f(x)=0
lim（x->0)f(x)/x=lim（x->0)f'(x)=0,所以f'(x)=0
设L=lim(x->0)(1+f(x)/x)^1/x
ln(L)=lim(x->0)ln(1+f(x)/x)/x
=lim[x->0)(xf'(x)-f(x)]/[x^2+xf(x)]
=lim(x->0)f''(x)/[2+f(x)/x+f'(x)]
=2
所以L=e^2