Question

lim（n→∞）（ntan1/n）^n方

Answer 1

lim（n→∞）（ntan1/n）^n方
lime^{nln[ntan(1/n)]}
lim [lnx+lntan(1/x)]/(1/x)
令t=1/x，t→0
=lim [-1/t+1/(sintcost)]/(t)
=lim (t-sintcost)/(t^2sintcost)
=lim (1-cos2t)/(3t^2)=lim 2(sint)^2/3t^2=2/3
e^（2/3）

Answer 2

=e^lim（n→∞）ln（ntan1/n）^n
=e^lim（n→∞）n·ln（ntan1/n）
=e^ lim（n→∞）n·ln(1+ ntan1/n -1)
=e^ lim（n→∞）n·[n·tan(1/n) -1]
=e^ lim（n→∞）[n·tan(1/n) -1]/(1/n)
=e^ lim（n→∞）[tan(1/n) - sec²(1/n) /n]/(-1/n²)
=e^ lim（n→∞）[0 - sec²(1/n) /n]/(-1/n²)
=e^ lim（n→∞）n·sec²(1/n)
→∞