Let $f:\mathbb{R} - \{-5\}->\mathbb{R}$, $f(x)=(x-1)e^{-(1/(x+5))}$. I have to calculate $lim_{(n->\infty)}=n^2\int_{0}^{1}x^nf(x)dx$.

I've tried using integration by parts, but i'm still stuck.


closed as off-topic by Carlo Beenakker, fedja, YCor, David Handelman, GH from MO Jul 11 at 22:36

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  • $\begingroup$ only the region near $x=1$ will contribute for large $n$, so you may replace $f(x)\mapsto (x-1)e^{-1/6}$ and find that the desired limit is $-e^{-1/6}$ --- I guess the question will be closed here, not research level math. $\endgroup$ – Carlo Beenakker Jul 11 at 21:08

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