Finding the Limit

Date: 1/25/96 at 21:54:43
From: Anonymous
Subject: Limit problem?

Hello Dr.Math!

I have a question about the limit of a sequence which I found it 
from my Calculus teacher :

  Proof: l i m   ( x^n / n! ) = 0 for any real value of x .
         n->infinity

I am frustrated about it and hope you can give any help!
I really appreciate it! Thank you very much!

  --George

Date: 1/26/96 at 16:27:7
From: Doctor Byron
Subject: Re: Limit problem?

Hello!

Here's how I'd think about this problem.  Let's say x is 2.  
Then as n gets really big, the limit will look like this:

2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2  x ...
--------------------------------------------
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x ...

Now, you notice that eventually all the top terms to the right of 
a certain point are smaller than the terms directly under them.  
That's because the factorial on the bottom keeps multiplying by 
bigger and bigger numbers, while the stuff on the top keeps only 
multiplying by 2.  Let's look at another example, where x is 173:

173 x 173 x 173 x ... x 173 x 173 x 173 x 173 x 173 x 173 x ...
---------------------------------------------------------------
 1  x  2  x  3  x ... x 172 x 173 x 174 x 175 x 176 x 177 x ...

Notice that it takes longer, but the bottom still gets bigger than 
the top.  So when the bottom of a fraction gets a lot bigger than 
the top, the whole fraction gets close to 0, which is why the limit 
is 0.

-Doctor Byron,  The Math Forum