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How can I show that $$ \lim_{A\rightarrow \infty} \int_0^A \frac{\sin(x)}{x}dx\;=\;\frac{\pi}{2}?$$ I know that can use the fact that, for $x>0$, $$x^{-1}\;=\;\int_0^\infty e^{-xt}dt$$ but I'm not sure how to begin.

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