lim(n→∞)(ntan1/n)^n方
2012-03-14
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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)
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)
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=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)
→∞
=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)
→∞
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