Question

设f(x)有二阶导数，在x=0的某去心邻域内f(x)≠0，且lim f(x)/x=0,f'(0)=4

求lim (1+f(x)/x)^(1/x)

Answer 1

由limf(x)/x=0得f'(0)=0

ln[1+f(x)/x]~x(x->0)
limln(1+f(x)/x)^(1/x)=limln[1+f(x)/x]/x=limf(x)/x^2=limf'(x)/2x=f''(0)/2=2
原式=e^2