Indeterminate Forms

Date: 04/23/2001 at 18:18:22
From: Brian Jennings
Subject: Indeterminate Forms

Concerning the indeterminate forms such as 0/0 and infinity/infinity, 
why is one to the infinite power considered an indeterminate form?

Date: 04/23/2001 at 18:42:14
From: Doctor Schwa
Subject: Re: Indeterminate Forms

Hi Brian,

I think of them not as "0/0" but as "tiny/tiny."  It's indeterminate 
because 0.000002/0.000001 is tiny/tiny that equals 2, but you could 
have all kinds of other things like 0.00000000000001/0.00000001, which 
is very close to 0, and so on (that is, you can have 2x/x or x^2/x or 
all sorts of other things).

Now, if you think of 1 to the infinite power as (very close to 1) to 
the (huge power), you can get things like .999999^1000000, which is 
very close to 1/e, or .999999^1000000000000, which is very close to 0, 
or 1.000001^1000000, which is very close to e, or 
1.000001^1000000000000, which is huge, or anything in between ...

That is,

         (1 - 1/x)^x -> 1/e
     (1 - 1/x)^(x^2) -> 0
         (1 + 1/x)^x -> e
     (1 + 1/x)^(x^2) -> infinity

and many other choices, in particular, (1 + n/x)^x -> e^n, so by 
choosing values of n you can get any positive value you want for the 
limit.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/