求极限lim_x→∞[sin(1/x)+cosx(1/x)]^x

求极限lim_x→∞[sin(1/x)+cosx(1/x)]^x

这是一个1^∞未定式，变形后得 lim_x→∞(sin1/x+cos1/x)^x=lim_x→∞e^{xln(sin1/x+cos1/x)}=e^{lim_x→∞ln(sin1/x+cos1/x)/(1/x)} 令t=1/x，则x→∞⇔t→0并且 lim_{x→ ∞}ln(sin1/x+cosx1/x)/(1/x)=lim_t→0ln(sint+cost)/t =lim_t→0[ln(sint+cost)]′/t′=lim_t→0(cost-sint)/(sint+cost)=1 所以lim_t→0(sin1/x+cos1/x)^x=e．