# Thread: One to the power of infinity

1. ## One to the power of infinity

Hi,

I don't understand why one to the power of infinity isn't equals to one.

Pls enlighten me. Thank you

2. Originally Posted by mathbuoy
Hi,

I don't understand why one to the power of infinity isn't equals to one.

Pls enlighten me. Thank you
What are you told it equals?

We have

$\displaystyle \displaystyle \lim_{x \to \infty} 1^x = 1$

Not sure what else 1 to the power of infinity is supposed to mean.

3. Originally Posted by undefined
What are you told it equals?

We have

$\displaystyle \displaystyle \lim_{x \to \infty} 1^x = 1$

Not sure what else 1 to the power of infinity is supposed to mean.
2.718.. something.

4. Originally Posted by mathbuoy
2.718.. something.
$\displaystyle \displaystyle \lim_{x \to \infty}\left(1+\dfrac{1}{x} \right)^x = e \approx 2.71828$

5. Originally Posted by undefined
$\displaystyle \displaystyle \lim_{x \to \infty}\left(1+\dfrac{1}{x} \right)^x = e \approx 2.71828$
But of course that is not one to infinity which is undefined. Since we are talking limits and we can make

$\displaystyle \displaystyle \lim_{x \to \infty} [f(x)]^x$

equal to any positive quantity (and even infinite) for some choice of $\displaystyle$$f(x)$ for which:

$\displaystyle \displaystyle \lim_{x \to \infty} f(x)=1$

CB

6. Originally Posted by mathbuoy
2.718.. something.
Did this happen.... ??

$\displaystyle \displaystyle \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$

As n approaches infinity, $\displaystyle \frac{1}{n}\ \rightarrow\ 0$

leaving 1.

Unfortunately, that would be ok if the expression in brackets was not raised to the value n, which is approaching infinity.
The power "n" prevents us from making that internal simplification.

7. I'd like to point out that I mentioned this

$\displaystyle \displaystyle \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$

because it is a very well known limit whose value matches what the OP said was expected, not because I thought it was related to 1 to the power of infinity. I didn't explain because I was expecting the OP to respond something like, "Oh yeah that's the limit, I made a typo" or "No that's unrelated, I meant something else..." etc.

CaptainBlack pointed out something that reminded me of L'hopital's rule, and Archie Meade pointed out a possible manipulation error that could lead to a confusion of these limits, which are both useful insights I think.

8. I suspect that the OP was looking at

$\displaystyle \displaystyle \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$

and CaptainBlack's post applies; see this thread.