Compute the limit$$\lim_{k \rightarrow \infty } \int_0^k x^{-k} e^{x^2/k^2}\sin(x/k)~\mathrm dx.$$
Completely stuck with this one. Some convergence theorem is obviously needed, but can't figure out which one and how to apply it.
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1$\begingroup$ Are you sure the integral is written correctly? We have $x^{-k} e^{x^2/k^2}\sin(x/k)\sim \frac{1}{k}x^{-k+1}$ as $x\to 0^+$ and $x^{-k+1}$ has a non-integrable singularity at $0$ for $k\geq 2$. $\endgroup$– Alann RosasCommented Jun 18 at 7:21
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$\begingroup$ This is the way it is presented in the set of exercises I'm working on. There could be a mistake as I found basically the same exercise where $x^{-k}$ is replaced with $x^{-2}$ and the integral goes from 1 to infinity. This seems to make much more sense. $\endgroup$– Anonymous11Commented Jun 18 at 7:50
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