Question: I'm trying to evaluate
$\lim\_{e\to 0} \, \frac{i}{e}\int\_{\pi }^0 \frac{1-\exp (i e \exp (i \theta ))}{\exp (i \theta )} \, d\theta$ with Mathematica 9.0.1.0 on OS X.

However, I get "Undefined" for this input:

```
Limit[I/e Integrate[(1 - Exp[I e Exp[I θ]])/Exp[I θ], {θ, π, 0}], e -> 0]

```

We can see numerically that the correct answer is $-\pi$ by using this code:

```
With[{e = 0.000001}, I/e NIntegrate[(1 - Exp[I e Exp[I θ]])/Exp[I θ], {θ, π, 0}] ] // Chop

```

What goes wrong in this process? Is it possible to get the correct result?

Answer: 
```
Assuming[s \[Element] Reals && s > 0, 
 Limit[I/s Integrate[(1 - Exp[I s Exp[I t]])/Exp[I t], {t, Pi, 0}], 
  s -> 0]]

```

yields $-\pi$ (Mathematica 9.0). The integral yields a conditional expression which renders limit undefined without declaring assumptions. There is a difficulty (probably related to periodicity) if you try to bypass by using generate conditions to false. The limit returned is $\pi$.
You can use [`AsymptoticIntegrate`](http://reference.wolfram.com/language/ref/AsymptoticIntegrate):

```
I/e AsymptoticIntegrate[(1 - Exp[I e Exp[I θ]])/Exp[I θ, {θ, π, 0}, {e, 0, 1}]

```

> 
> -π
> 
> 
>