设函数f（x）在x=12的某邻域内可导，且limx→12f（x）=0，limx→12f′（x）=A，则limx→12∫x12[t•∫12tf(θ)dθ]dt(12-x)3=．

Question

设函数f（x）在x=12的某邻域内可导，且

lim

x→12

f（x）=0，

lim

x→12

f′（x）=A，则

lim

x→12

(∫x12[t•∫12tf(θ)dθ]dt)/((12-x)3)=___．

Answer 1

lim

x→12

(∫x12[t•∫12tf(θ)dθ]dt)/((12-x)3)
(洛必达法则.)/()

lim

x→12

(x∫12xf(θ)dθ)/(3(12-x)2)
=-

lim

x→12

(x∫x12f(θ)dθ)/(3(x-12)2)
(洛必达法则.)/()-

lim

x→12

(∫x12f(θ)dθ-xf(x))/(6(12-x))
(洛必达法则.)/() 

lim

x→12

(xf′(x))/(6)
=2

lim

x→12

f′(x)
=2A．
故答案为：12．