Limiting a Complex Function to π

[rondo09]

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

 

[HallsofIvy]

First, of course, that "x" outside the integral goes to $\pi$. The only problem is
$$\lim_{x\to\pi}\frac{\int_\pi^x \frac{sin(t)}{t} dt}{x- \pi}$$
which gives the "0/0" indeterminate form.

Use L'Hopital's rule to find that limit.

 

[mathman]

First, of course, that "x" outside the integral goes to $\pi$. The only problem is
$$\lim_{x\to\pi}\frac{\int_\pi^x \frac{sin(t)}{t} dt}{x- \pi}$$
which gives the "0/0" indeterminate form.

Use L'Hopital's rule to find that limit.

The expression is simply the derivative of the integral, i.e. the integrand at π, which is sin(π)/π = 0.