$\displaystyle {{\lim_{\substack{x\rightarrow\pi}} {\left( \frac {x}{x-\pi}{\int_{\pi}^{x} }\frac{sin t}{t}} dt\right)}$

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- Aug 1st 2010, 02:51 AMrondo09Real analysis
$\displaystyle {{\lim_{\substack{x\rightarrow\pi}} {\left( \frac {x}{x-\pi}{\int_{\pi}^{x} }\frac{sin t}{t}} dt\right)}$

- Aug 1st 2010, 04:37 AMHallsofIvy
Of course, you can treat the "x" outside the integral separately- it goes to $\displaystyle \pi$. That leaves $\displaystyle \frac{\int_\pi^x\frac{sin(t)}{t}dt}{x- \pi}$ which is of the form "0/0". Use L'Hopital's rule for that.