1.原式=lim(x->0)∫(0,x)ln(cost)dt/x³
=lim(x->0)[ln(cosx)/3x²] (0/0型,应用一次罗比达法则)
=lim(x->0)[(-sinx/cosx)/6x] (0/0型,应用再一次罗比达法则)
=lim(x->0)[(sinx/x)*(-1/6cosx)]
=lim(x->0)[(sinx/x)*lim(x->0)(-1/6cosx)
=1*(-1/6)
=-1/6.
2.原式=∫(-1,3)(|2-x|)dx
=∫(-1,0)(x-2)dx+∫(0,3)(2-x)dx
=(x²/2-2x)|(-1,0)+(2x-x²/2)|(0,3)
=(-1/2-2)+(6-9/2)
=-1.
3.两个积分相等,
即∫(0,1)dy∫(0,√y)e^yf(x)dx=∫(0,1)dx∫(x²,1)e^yf(x)dy.
4.∵级数∑（下n=1上趋于无穷）Un收敛
∴根据级数收敛的必要条件,有lim(n趋于无穷）Un=0.