1.∵左极限=lim(x->0-)[ln(1-x)/(-x(x+3))]
=lim(x->0-)[1/(x+3)]*lim(x->0-)[ln(1-x)/(-x)]
=(1/3)lim(x->0-)[ln(1-x)/(-x)]
=(1/3)lim(x->0-)[(-1/(1-x))/(-1)] (0/0型极限,应用洛必达法则)
=1/3；
右极限=lim(x->0+)[ln(1-x)/(x(x+3))]
=lim(x->0+)[1/(x+3)]*lim(x->0+)[ln(1-x)/x]
=(1/3)lim(x->0+)[ln(1-x)/x]
=(1/3)lim(x->0+)[-1/(1-x)] (0/0型极限,应用洛必达法则)
=-1/3.
∴根据间断点分类定义知,x=0是函数f(x)=ln|1-x|/|x|(x+3)的第一类间断点.
2.∵f(x)=2x²lnx
∴它的定义域是(0,+∞)
∵令f'(x)=2(2xlnx+x)=0,得x=e^(-1/2)
当00.即此函数严格单调递增.
∴f(x)=2x²lnx的单调增区间为(e^(-1/2),+∞).
3.∵f(x)有连续的二阶导数,且f'(b)=a,f'(a)=b
∴所求积分存在
故∫f'(x)f''(x)dx=∫f'(x)d(f'(x))
=[(f'(x))²/2]│
=[(f'(b))²-(f'(a))²]/2
=(a²-b²)/2.