求极限limx→∞[sin(1/x)+cos(1/x)]2x
【正确答案】：limx→∞[sin(1/x)+cos(1/x)]2x=limx→∞e2xln[sin(1/x)+cos(1/x)] =elimx→∞2xln[sin(1/x)+cos(1/x)]， 而limx→∞2xln[sin(1/x)+cos(1/x)]=2limx→∞{ln[sin(1/x)+cos(1/x)]/(1/x)}=2 limt→0[ln(sint+cost)/t] (t=1/x) =2limt→0{[ln(sint+cost)]′/(t)′}=2limt→0[(cost-sint)/(sint+cost)]=2. 所以原极限=e2．
求极限limx→∞[sin(1/x)+cos(1/x)]2x
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