Question

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

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