Indeterminate FormsDate: 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/ |
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