Evaluating expression at infinity

Question

How do I evaluate something like:

$$xe^{-(x-\theta)}\text{ from }x = \theta\text{ to }x=\infty?$$

This came up in an integration I tried to do, and I realize it's a very basic question. But I am confused on how to properly evaluate the expression at $\infty$.

Thanks in advance.

Answer 1

Obviously when $x=\theta$ we get $$\theta e^{-(\theta - \theta)} = \theta \cdot e^0 = \theta \cdot 1 = \theta$$

For the other infinite case, you should know that the exponential grows faster than the linear function, so $e^{-(x-\theta)}$ goes to zero faster than $x$ goes to infinity, hence:

$$\lim_{x \to \infty} xe^{-(x-\theta)} = 0$$

Answer 2

You take the limit, so:

$$ xe^{-(x-\theta)}\,\Big|_\theta^\infty=\left(\lim_{x\to\infty}xe^{-(x-\theta)}\right)-\theta e^{-(\theta-\theta)}=0-\theta=\theta.$$