求limx→∞[(3x+2)/(3x-1)]^2x-1

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求limx→∞[(3x+2)/(3x-1)]^2x-1

求limx→∞[(3x+2)/(3x-1)]^2x-1
求limx→∞[(3x+2)/(3x-1)]^2x-1

求limx→∞[(3x+2)/(3x-1)]^2x-1
运用重要极限 lim(x→0)(1+x)^(1/x)=e
lim(x→∞)[(3x+2)/(3x-1)]^2x-1
=lim(x→∞)[1+3/(3x-1)]^[(3x-1)/3]*[3*(2x-1)/(3x-1)]
=e^lim(x→∞)[3*(2x-1)/(3x-1)]
=e^2

[(3x+2)/(3x-1)]^2x-1=[1+1/(x-1/3]^2x-1
令t=x-1/3
那么x=t+1/3
那么
原式=(1+1/t)^(2t-1/3)=(1+1/t)^2t *(1+1/t)^(-1/3)
x趋近于无穷,那么t也趋近于无穷
那么原式=e^2 *1=e^2