求极限limx→∞[sin(1/x)+cos(1/x)]2x
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.