n趋近无穷大,n的1/n次幂的极限为什么等于一

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n趋近无穷大,n的1/n次幂的极限为什么等于一n趋近无穷大,n的1/n次幂的极限为什么等于一n趋近无穷大,n的1/n次幂的极限为什么等于一设limx^(1/x)=AlnA=lim(1/x)*lnx=l

n趋近无穷大,n的1/n次幂的极限为什么等于一
n趋近无穷大,n的1/n次幂的极限为什么等于一

n趋近无穷大,n的1/n次幂的极限为什么等于一
设lim x^(1/x)=A
lnA=lim (1/x)*lnx=lim lnx/x
lnA=lim (1/x)/1=0（求导)
A=1

本来就是趋向于1
lim n^1/n=lim n的根号n次方
趋向于1

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可以进行理论证明，但也可直接观察，你看一下面直接计算所得的数据，每组中第一个数是n的取值，第二个就是n的1/n次幂。
{100, 1.04713}, {200, 1.02685}, {300, 1.01919}, {400, 1.01509}, {500,
1.01251}, {600, 1.01072}, {700, 1.0094}, {800, 1.00839}, {900,
1.00759}, {1000, 1.00693}, {1100, 1.00639}, {1200, 1.00593}, {1300,
1.00553}, {1400, 1.00519}, {1500, 1.00489}, {1600, 1.00462}, {1700,
1.00439}, {1800, 1.00417}, {1900, 1.00398}, {2000, 1.00381}, {2100,
1.00365}, {2200, 1.0035}, {2300, 1.00337}, {2400, 1.00325}, {2500,
1.00313}, {2600, 1.00303}, {2700, 1.00293}, {2800, 1.00284}, {2900,
1.00275}, {3000, 1.00267}, {3100, 1.0026}, {3200, 1.00253}, {3300,
1.00246}, {3400, 1.00239}, {3500, 1.00233}, {3600, 1.00228}, {3700,
1.00222}, {3800, 1.00217}, {3900, 1.00212}, {4000, 1.00208}, {4100,
1.00203}, {4200, 1.00199}, {4300, 1.00195}, {4400, 1.00191}, {4500,
1.00187}, {4600, 1.00184}, {4700, 1.0018}, {4800, 1.00177}, {4900,
1.00174}, {5000, 1.0017}, {5100, 1.00168}, {5200, 1.00165}, {5300,
1.00162}, {5400, 1.00159}, {5500, 1.00157}, {5600, 1.00154}, {5700,
1.00152}, {5800, 1.0015}, {5900, 1.00147}, {6000, 1.00145}, {6100,
1.00143}, {6200, 1.00141}, {6300, 1.00139}, {6400, 1.00137}, {6500,
1.00135}, {6600, 1.00133}, {6700, 1.00132}, {6800, 1.0013}, {6900,
1.00128}, {7000, 1.00127}, {7100, 1.00125}, {7200, 1.00123}, {7300,
1.00122}, {7400, 1.0012}, {7500, 1.00119}, {7600, 1.00118}, {7700,
1.00116}, {7800, 1.00115}, {7900, 1.00114}, {8000, 1.00112}, {8100,
1.00111}, {8200, 1.0011}, {8300, 1.00109}, {8400, 1.00108}, {8500,
1.00107}, {8600, 1.00105}, {8700, 1.00104}, {8800, 1.00103}, {8900,
1.00102}, {9000, 1.00101}, {9100, 1.001}, {9200, 1.00099}, {9300,
1.00098}, {9400, 1.00097}, {9500, 1.00096}, {9600, 1.00096}, {9700,
1.00095}, {9800, 1.00094}, {9900, 1.00093}, {10000, 1.00092}
不知道你是否相信这个事实了。

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