I'm stuck with this limit $(1 - \frac{c}{n}\log n )^{1-n}$ as $n \rightarrow \infty$ where $c < 1$. I tried to plot the limit and it looks like it goes to infinity, although very slowly, but I can't prove it. Any ideas?
A Tricky Limit: $(1 - \frac{c}{n}\log n )^{1-n}$
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calculus
limits
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0Consider taking the limit of $(1-n)\log(1-(c\log n)/n)$... – 2011-10-10
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2[Wolfram Alpha](http://www.wolframalpha.com/input/?_=1318257578883&i=(1+-+%5cfrac%7bc%7d%7bn%7d%5clog+n+)%5e%7b1-n%7d+as+n+-%3e+infinity&fp=1&incTime=true) says, $\lim_{n\to\infty}(1-\frac cn\log n)^{1-n} = \infty$... – 2011-10-10
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1Hence the sequence is $n^{c+o(1)}$, which is slow but not so slow... – 2011-10-10
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1For $0
the my result came $\infty$. – 2011-10-10 -
0I tried writting the expression as : $L:=\lim_{n \to \infty} \exp{(1-n+\frac{(n-c)}{n} \log n)}$. – 2011-10-10
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0Oops.. I messed up the expression by missing $\log$. Read $L:=\lim_{n \to \infty} \exp{(1-n) \log {(\frac{(n-c \log n)}{n}}}$ – 2011-10-10