limit of integral (Riemann-Lebesgue lemma?)

[sonic meadow]

I tried to do it with the lemma in the second pic, but I’m unsure if that’s the right approach here, since the limit looks pretty undefined when t=0.

I also tried to write it in terms of the dirichlet kernel (because the chapter is about pointwise convergence for Fourier series) but idk where to go next. Any lead is appreciated!

[knotty locust]

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