Question

f(x)=(g(x)-e^(-x))/x,(x不等于0)；0（x=0)。g(x)有二阶连续导数,g(0)=1，g'(x)=-1..求f'(x)在R上的连续性

Answer 1

先证明f(x)在x=0可导
lim(x->0) [f(x)-f(0)]/x
=lim [g(x)-e^(-x)] / x^2
洛必达法则
=lim [g'(x)+e^(-x)] / (2x)
洛必达法则
=lim [g''(x)-e^(-x)]/2
=(g''(0)-1)/2

因为g(x)有二阶连续导数
所以x≠0时f'(x)=[x(g'(x)+e^(-x)) - (g(x)-e^(-x))] / x^2连续

再证明f‘(x)在x=0处连续
lim(x-0) f'(x)
=lim [x(g'(x)+e^(-x)) - (g(x)-e^(-x))] / x^2
洛必达法则
=lim [xg''(x) + g'(x) - xe^(-x) + e^(-x) - g'(x) - e^(-x)] / 2x
=lim [g''(x)-e^(-x)]/2
=[g''(0)-1]/2
所以f'(x)在x=0处连续
所以f'(x)在R上连续