求x趋于0时lim(1/x)积分符号(上1下0)f(xt)dtf(x)可导,连续,f(0)=0（答案是0.5f'(0),

Question

求x趋于0时lim(1/x)积分符号(上1下0)f(xt)dt
f(x)可导,连续,f(0)=0 （答案是0.5f'(0),

Answer 1

lim{x->0} (1/x) ∫[0, 1] f(xt) dt
= ∫[0, 1] t*lim{xt->0}{f(xt)-f(0)}/(xt) dt
= ∫[0, 1] t*f'(0) dt,注意：lim{xt->0}{f(xt)-f(0)}/(xt) = f'(0)
= f'(0)(0.5t^2) from 0 to 1
= 0.5f'(0)
美国高中数学老师