高等数学::::设lim[x→0][xf(x)+ln(1-2x)]/x^2=4,则lim[x→0][f(x)-2]/x=.请老师讲解一下,谢谢您了,答案是6,我不知道怎么来的

Question

高等数学::::
设lim[x→0] [xf(x)+ln(1-2x)] / x^2=4,则lim[x→0] [f(x)-2] /x =_____.
请老师讲解一下,谢谢您了,答案是6,我不知道怎么来的

Answer 1

lim[x→0] [xf(x)+ln(1-2x)] / x^2=lim[x→0] [xf(x)-2x+2x+ln(1-2x)] / x^2
=lim[x→0] [xf(x)-2x】/x^2+lim[x→0] [2x+ln(1-2x)] / x^2
=lim[x→0] [f(x)-x】/x+lim[x→0] [2-2/(1-2x)] / 2x
=lim[x→0] [f(x)-x】/x+lim[x→0] [-4x/(1-2x)* 2x]
=lim[x→0] [f(x)-x】/x+lim[x→0] [-2/(1-2x)]
=lim[x→0] [f(x)-x】/x-2=4
lim[x→0] [f(x)-2] /x =6