Fairly simple limit proof?

**scorpion007** · June 27th 2010, 03:57 AM

How would I prove that

$\lim\limits_{x\to\infty}xe^{-x}=0$ ?

I've only done eps-delta proofs for finite limit points, not to infinity.

**Deadstar** · June 27th 2010, 04:18 AM

Are you familiar with L'Hopitals rule?

**scorpion007** · June 27th 2010, 04:29 AM

Yes. I think I may know how to solve it then. -- Wait, but LHopitals rule is for finding the limit, not proving it?

**mr fantastic** · June 27th 2010, 05:20 AM

Originally Posted by scorpion007

Yes. I think I may know how to solve it then. -- Wait, but LHopitals rule is for finding the limit, not proving it?

Surely if you find that the limit is equal to zero by using a valid theorem like l'Hopital's Rule, then you have proved that the limit is equal to zero! But context might be important ..... Where has the question come from?

**scorpion007** · June 27th 2010, 04:18 PM

It arose indirectly from me solving an integral of the form $\int_0^\infty xe^{-x}$ where I had to evaluate the limit of a similar expression, and was curious if I could prove it using a method like e-d proofs.

**Plato** · June 27th 2010, 05:09 PM

Originally Posted by scorpion007

It arose indirectly from me solving an integral of the form $\int_0^\infty xe^{-x}$ where I had to evaluate the limit of a similar expression, and was curious if I could prove it using a method like e-d proofs.

In that case, why use a cannon when a flyswatter will do?

**scorpion007** · June 27th 2010, 05:19 PM

Only because I was curious if I could prove it -- not because it was strictly necessary to do so.

**Also sprach Zarathustra** · June 27th 2010, 05:28 PM

Originally Posted by scorpion007

How would I prove that

$\lim\limits_{x\to\infty}xe^{-x}=0$ ?

I've only done eps-delta proofs for finite limit points, not to infinity.

A some kind of try:

By definition...

For every $\epsilon>0$ exist $x>1$ so that:

$|xe^{-x}|<\frac{1}{x}<\epsilon$

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