how to evaluate lim_{n->infinity} int_0^1 e^{x^2} sin(nx) dx ?

11. The value of [imath]\displaystyle \lim_{n \rarr \infty} \int_0^1 e^{x^2}\, \sin(nx)\, dx[/math] is ______? [/QUOTE] What are your thoughts? What have you tried? How far have you gotten? ("[URL='https://www.freemathhelp.com/forum/threads/read-before-posting.41538/']Read Before Posting[/URL]") Please be complete. Thank you![/imath]
 
firstly i tried with the gamma function but it is not getting anywhere with that
 
I'd try using 01ex2sin(nx)dx=O(1n)+k2πkn2π(k+1)nex2sinnx  dx\int_0^1 e^{x^2} \sin (nx) dx = O\left(\frac{1}{n}\right) + \sum_k \int_{\frac{2\pi k}{n}}^{\frac{2\pi(k+1)}{n}} e^{x^2} \sin nx\;dx then replace u=nxu=nx and use the fact that ex2e^{x^2} has limited derivativet to show that each summand is of order O(1n2)O\left(\frac{1}{n^2}\right)
 
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