抱歉啊,昨天不好意思把e漏掉了,多亏你提醒。现在我已改了,而且还漏了"-"号,原答案中"-"号去掉:
1.此为无穷*0型,先改写成0/0型,再用洛比达法则:
=lim[e-(1+1/n)^n]/(1/n) n→+∞
分子改写成e指数:
=lim {e-e^[nln(1+1/n)]}/(1/n)
由洛比达法则:
=lim (-e^[nln(1+1/n)])[ln(1+1/n)-1/(n+1)]/(-1/n²)
=lim (-e)[ln(1+1/n)-1/(n+1)]/(-1/n²)
=lim e[ln(1+1/n)-1/(n+1)]/(1/n²)
由洛比达法则:
=lim e[-1/n(n+1)+1/(n+1)²]/(-2/n³)
=lim e[-1/n(n+1)²]/(-2/n³)
=lim e[n³/2n(n+1)²]
=e/2
2.此题用等价无穷小代换:
a^x-1≈xlna,x→0
原极限=n[n^(1/n)-1] n→+∞
=n[(1/n)lnn]
=lnn
=+∞
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