f(x)在点X=1处可导,f'(1)=1且则Limf(1-2h)-f(1+h)/h的值,最好写出计算过程,谢谢

Question

f(x)在点X=1处可导,f'(1)=1且则Limf(1-2h)-f(1+h)/h的值,最好写出计算过程,谢谢

Answer 1

Lim(h->0) f(1-2h)-f(1+h)/h
=Lim(h->0) [f(1-2h)-f(1)+f(1)-f(1+h)] /h
=Lim(h->0) [f(1-2h)-f(1)]/h - [f(1+h)-f(1)] /h
显然由导数的定义可以知道,
Lim(h->0) [f(1+h)-f(1)] /h =f '(1)
而
Lim(h->0) [f(1-2h)-f(1)]/h
=Lim(h->0) -2 [f(1)- f(1-2h)] / 2h
= -2f '(1)
故
Lim(h->0) f(1-2h)-f(1+h)/h
=Lim(h->0) [f(1-2h)-f(1)]/h - [f(1+h)-f(1)] /h
= -3f '(1)
= -3