当x趋近1时求{(x^1/2-1)(x^1/3-1).(x^1/n-1)}/(x-1)^(n-1)的极限

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当x趋近1时求{(x^1/2-1)(x^1/3-1).(x^1/n-1)}/(x-1)^(n-1)的极限
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当x趋近1时求{(x^1/2-1)(x^1/3-1).(x^1/n-1)}/(x-1)^(n-1)的极限
当x趋近1时求{(x^1/2-1)(x^1/3-1).(x^1/n-1)}/(x-1)^(n-1)的极限

当x趋近1时求{(x^1/2-1)(x^1/3-1).(x^1/n-1)}/(x-1)^(n-1)的极限
x^1/k-1=(x-1)/[1+x^1/k+x^2/k+...+x^(k-1)/k],k=2,3,...n
lim{(x^1/2-1)(x^1/3-1).(x^1/n-1)}/(x-1)^(n-1)
=1/(2*3*...(n))=1/n)!,n=2,3,4...