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Lebesgue Measures and Integrals : Let a,b be real numbers such that 0 < a < b < infinity. Does the limit [lim of ( integral from a to b of n*sin (x^2/n) dx], n is positive integer exist?

Let a,b be real numbers such that 0 < a < b < infinity. Does the limit

lim of ( integral from a to b of n*sin (x^2/n) dx , n is positive integer

exist? ( prove or disprove). Find the limit if it exists. Prove all assertions and justify every step.
The integral here is with respect of Lebesgue measure.

Solution Summary

Lebesgue Measures, Limits and Integrals are investigated. The solution is detailed and well presented.

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