求极限 lim(x趋向无穷)[(x+1)^(x+1)/x^x]sin(1/x)

求极限 lim(x趋向无穷)[(x+1)^(x+1)/x^x]sin(1/x)
求极限 lim(x趋向无穷)[(x+1)^(x+1)/x^x]sin(1/x)

求极限 lim(x趋向无穷)[(x+1)^(x+1)/x^x]sin(1/x)
sin（1/x)~1/x,
所以那个式子就化成了（1+1/x）^(x+1),
然后化为指数函数,用洛必达法则,这样楼主懂了呗?
不会的话在追问,

得lim(x趋向无穷)[(x+1)^(x+1)/x^x]1／x＝lim(x趋向无穷)[(x+1)^(x+1)/x^﹙x＋1﹚＝∞

lim[x→∝] [(x+1)^(x+1)/x^x]sin(1/x)
= lim[x→∝] [(x+1)^(x+1)/x^x]*(1/x)
= lim[x→∝] (x+1)^(x+1)/x^(x+1)
= lim[x→∝] [(x+1)/x]^(x+1)
= lim[x→∝] (1+1/x)^(x+1)
= lim[x→∝] e^ln[(1+1/x)^(x+...

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lim[x→∝] [(x+1)^(x+1)/x^x]sin(1/x)
= lim[x→∝] [(x+1)^(x+1)/x^x]*(1/x)
= lim[x→∝] (x+1)^(x+1)/x^(x+1)
= lim[x→∝] [(x+1)/x]^(x+1)
= lim[x→∝] (1+1/x)^(x+1)
= lim[x→∝] e^ln[(1+1/x)^(x+1)]
= e^lim[x→∝] (x+1)ln(1+1/x)
= e^lim[x→∝] ln(1+1/x)/[1/(x+1)]
= e^lim[x→∝] [-1/x^2(1+1/x)]/[-1/(x+1)^2]
= e^lim[x→∝] (x+1)/x
= e^1
= e

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