求极限limx→∞[sin(1/x)+cosx(1/x)]x
【正确答案】：这是一个1∞未定式，变形后得 limx→∞(sin1/x+cos1/x)x=limx→∞exln(sin1/x+cos1/x)=elimx→∞ln(sin1/x+cos1/x)/(1/x) 令t=1/x，则x→∞⇔t→0并且 limx→ ∞ln(sin1/x+cosx1/x)/(1/x)=limt→0ln(sint+cost)/t =limt→0[ln(sint+cost)]′/t′=limt→0(cost-sint)/(sint+cost)=1 所以limt→0(sin1/x+cos1/x)x=e．
