设f（x）=2x2(1−cosx)，x＜01，x＝01x∫x0cost2dt，x＞0．试讨论f（x）在x=0处的连续性和可导性．

Question

设f（x）=

(2)/(x2)(1−cosx)，  x＜0

1，    x＝0

(1)/(x)

∫

x 0

cost2dt，  x＞0

．试讨论f（x）在x=0处的连续性和可导性．

Answer 1

∵f′−(0)＝limx→0−f(x)−f(0)x=limx→0−2x2(1−cosx)−1x=limx→0−2(1−cosx)−x2x3=limx→0−2sinx−2x3x2＝limx→0−2cosx−26x═limx→0−−2sinx6＝0f′+(0)＝limx→0+f(x)−f(0)x=limx→0+1x∫x0cost2dt...