Question

已知f(x)在x=0处可导,且f(0)=0,则x趋近0时 lim(x^2f(x)-f(2x^3))/x^3=?

Answer 1

x趋近0时 lim(x^2f(x)-f(2x^3))/x^3
=lim(x->0)f(x)/x-lim(x->0)f(2x^3)/x^3
=lim(x->0)[f(x)-f(0)]/x-lim(x->0)[f(2x^3)-f(0)]/x^3
=f'(0)-2 lim(x->0)[f(2x^3)-f(0)]/2x^3
=f'(0)-2f'(0)
=-f'(0)

追答

不能直接使用洛必达

追问

lim(x->0)[f(x)-f(0)]/x=f'(0)是怎么来的？

追答

导数定义

Answer 2

∵ f(x)在x=0处可导,且f(0)=0
∴ 采用罗必塔法则
lim(x→0)[x²f(x)-f(2x³)]/x³
=lim(x→0)[f(x)/x]-lim(x→0)[f(2x³)/(x³)]
=lim(x→0)[f'(x)/1]-lim(x→0)[6x²f'(2x³)/(3x²)]
=f'(0)-lim(x→0)[2f'(2x³)]
=f'(0)-2f'(0)
=-f'(0)