lim[(1+1/x)^x^2]/e^x (X趋于正无穷)

5个回答

  • 先取自然对数得

    lim(x→∞)ln{[(1+1/x)^x^2]/e^x }

    =lim(x→∞)ln[(1+1/x)^x^2]-lne^x

    =lim(x→∞)x^2ln(1+1/x)-x (令x=1/t)

    =lim(t→0)ln(1+t)/t^2-1/t

    =lim(t→0)[ln(1+t)-t]/t^2 (运用洛必达法则)

    =lim(t→0)[1/(1+t)-1]/(2t)

    =lim(t→0)[-t/(1+t)]/(2t)

    =lim(t→0)-1/[2(1+t)]

    =-1/2

    所以

    lim(x→∞)[(1+1/x)^x^2]/e^x

    =lim(x→∞)e^ln{[(1+1/x)^x^2]/e^x }

    =e^(-1/2)