# Limit of zero to the zero

• Aug 21st 2012, 06:58 AM
VinceW
Limit of zero to the zero
$\lim\limits_{n \to \infty} \left(\frac{1}{n}\right)^\frac{1}{n}$. The answer is 1, but how do I show this?

My textbook says for this scenario take $y = f(x)^{g(x)}$, take $\ln y$, find the limit of that, and then the limit of $y = e^{\ln y}$. I'm not sure how to do this one either: $\lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right]$

EDIT: OK solved.

$\lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right]$ $= \lim\limits_{n \to \infty} \frac{\ln \frac{1}{n}}{n}$ which is solved with l'hopital's rule, comes to zero, and the total limit comes to one. thanks!
• Aug 25th 2012, 08:01 PM
topsquark
Re: Limit of zero to the zero
Quote:

Originally Posted by VinceW
$\lim\limits_{n \to \infty} \left(\frac{1}{n}\right)^\frac{1}{n}$. The answer is 1, but how do I show this?

My textbook says for this scenario take $y = f(x)^{g(x)}$, take $\ln y$, find the limit of that, and then the limit of $y = e^{\ln y}$. I'm not sure how to do this one either: $\lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right]$

EDIT: OK solved.

$\lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right]$ $= \lim\limits_{n \to \infty} \frac{\ln \frac{1}{n}}{n}$ which is solved with l'hopital's rule, comes to zero, and the total limit comes to one. thanks!

There is as much material to suggest that $0^0 = 0$. You have done the limit correctly (as far as I can see) but the problem $0^0$ has been labelled as undefined.

-Dan
• Aug 25th 2012, 11:52 PM
JJacquelin
Re: Limit of zero to the zero
Hi VinceW !

Just have a look at this paper and you will see several commented examples of 0^0 :
"Zéro puissance zéro - Zero to the zero power"
Scribd