求极限lim_{x→∞[sin(1/x)+cos(1/x)]^2x}

求极限lim_{x→∞[sin(1/x)+cos(1/x)]^2x}

lim_x→∞[sin(1/x)+cos(1/x)]^2x=lim_x→∞e^{2xln[sin(1/x)+cos(1/x)]} =e^{lim_x→∞2xln[sin(1/x)+cos(1/x)]}， 而lim_x→∞2xln[sin(1/x)+cos(1/x)]=2lim_x→∞{ln[sin(1/x)+cos(1/x)]/(1/x)}=2 lim_t→0[ln(sint+cost)/t] (t=1/x) =2lim_t→0{[ln(sint+cost)]′/(t)′}=2lim_t→0[(cost-sint)/(sint+cost)]=2. 所以原极限=e²．