F(x)=f(x)(1+|sinx|),F(x),f(x)在x=0处可导,求f(0)

问题描述:

F(x)=f(x)(1+|sinx|),F(x),f(x)在x=0处可导,求f(0)

F'(0)=lim(x->0) (f(x)(1+|sinx|)-f(0))/x-0=f'(x)(可导)+lim(x->0) f(x)|sinx|/x
lim(x->0) f(x)|sinx|/x 左极限为-f(0),右极限为f(0)
F(x)在0处可导,则左右极限相等,-f(0)=f(0),f(0)=0

F'(0)=lim{x->0}[f(x)(1+|sinx|)-f(0)]/x=lim{x->0}[f(x)-f(0)](1+|sinx|)/x+lim{x->0}[f(0)(1+|sinx|)-f(0)]/x=f'(0)+lim{x->0}f(0)|sinx|/x左极限lim{x->0-}f(0)|sinx|/x=-f(0),右极限lim{x->0+}f(0)|sinx|/x=f(0)...