求(1+(n-1)/n2)^(n^2+2n-2)/(n-1)(n趋于无穷大)的极限
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求(1+(n-1)/n2)^(n^2+2n-2)/(n-1)(n趋于无穷大)的极限
求(1+(n-1)/n2)^(n^2+2n-2)/(n-1)(n趋于无穷大)的极限
求(1+(n-1)/n2)^(n^2+2n-2)/(n-1)(n趋于无穷大)的极限
lim(1+(n-1)/n2)^(n^2+2n-2)/(n-1)
=e^(lim(n-1)/n2)*(n^2+2n-2)/(n-1))
=e^(lim(n^2+2n-2)/n^2)
=e^1
=e
n->∞,(n-1)/n^2->0
[1+(n-1)/n^2]^[(n^2+2n-2)/(n-1)]
=[1+(n-1)/n^2]^[n^2/(n-1) +2]
=[1+(n-1)/n^2]^n^2/(n-1) *[1+(n-1)/n^2]^2
n->∞,[1+(n-1)/n^2]^n^2/(n-1)->e,[1+(n-1)/n^2]^2->0
所以,当n->∞时 [1+(n-1)/n^2]^[(n^2+2n-2)/(n-1)]->e
求极限x→∞lim[1+(n-1)/n²)]^[(n²+2n-2)/(n-1)]
原式=x→∞lim[1+(n-1)/n²]^[(n+3)+1/(n-1)]
={x→∞lim[1+(n-1)/n²]^(n+3)}{x→∞lim[1+(n-1)/n²]^[1/(n-1)]}
={x→∞lim[1+(n-1)/n²]^...
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求极限x→∞lim[1+(n-1)/n²)]^[(n²+2n-2)/(n-1)]
原式=x→∞lim[1+(n-1)/n²]^[(n+3)+1/(n-1)]
={x→∞lim[1+(n-1)/n²]^(n+3)}{x→∞lim[1+(n-1)/n²]^[1/(n-1)]}
={x→∞lim[1+(n-1)/n²]^n}{{x→∞lim[1+(n-1)/n²]^3}{x→∞lim[1+(n-1)/n²]^[1/(n-1)]}
=x→∞lim[1+(n-1)/n²]^{[n²/(n+1)][(n+1)/n]}{x→∞lim[1+(n-1)/n²]^3}{x→∞lim[1+(n-1)/n²]^[1/(n-1)]}
其中x→∞lim[1+(n-1)/n²]^{[n²/(n+1)][(n+1)/n]}=e
x→∞lim[1+(n-1)/n²]^3=1
x→∞lim[1+(n-1)/n²]^[1/(n-1)]=x→∞lim[1+(n-1)/n²]^{[n²/(n-1)](1/n²)}=eº=1
故原式=e
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