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How does one prove the following limit? $$ \lim_{b\rightarrow 0}\int_0^\infty{\frac{\sin x}{x}e^{-bx}dx} = \lim_{N\rightarrow\infty}\int_0^N{\frac{\sin x}{x}dx} $$

I don't think I could apply the Dominated Convergence Theorem outright because any estimate I can come up with is not $L^1$, so I guess this becomes a question of commuting limits.

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