((x-1)/(1+x))^x
=e^(x*ln((x-1)/(x+1)))
当x趋于无穷时求((x-1)/(1+x))^x的极限等价于求e^(x*ln((x-1)/(x+1)))的极限
∴lim(x→∞)((x-1)/(1+x))^x
=lim(x→∞)(e^(x*ln((x-1)/(x+1))))
=e^(lim(x→∞)(x*ln((x-1)/(x+1))))
而lim(x→∞)(x*ln((x-1)/(x+1)))
=lim(x→∞)(ln((x-1)/(x+1))/1/x)
=lim(x→∞)(-2x²/(x²-1))
=-2
∴lim(x→∞)((x-1)/(1+x))^x
=lim(x→∞)e^(x*ln((x-1)/(x+1)))
=e^(-2)