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The task is to evaluate: $\displaystyle \lim_{n \to +\infty}\int_{n}^{n+7}\frac{\sin x}{x}\mbox{d}x$.

I don't know how to approach this. $\displaystyle \int_{}^{}\frac{\sin x}{x}\mbox{d}x$ doesn't even express in elementary functions.

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How about Taylor series? – The Chaz 2.0 May 4 '12 at 15:26
Isn't the function getting awfully close to $0$? And the interval is not getting long. – André Nicolas May 4 '12 at 15:34
up vote 5 down vote accepted

Hint: use a comparison test $\bigl|\,\int_n^{n+7}{\sin x\over x}\,dx\,\bigr|\le \int_n^{n+7} \bigl|{\sin x\over x}\bigr|\,dx \le\int_n^{n+7} {1\over x} \,dx$.

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