Limit of zero to the zero

**VinceW** · August 21st 2012, 05:58 AM

$\lim\limits_{n \to \infty} \left(\frac{1}{n}\right)^\frac{1}{n}$ . 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!

**topsquark** · August 25th 2012, 07:01 PM

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

**JJacquelin** · August 25th 2012, 10:52 PM

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

Thread: Limit of zero to the zero

LinkBack

Thread Tools

Display

Limit of zero to the zero

Re: Limit of zero to the zero

Re: Limit of zero to the zero

Similar Math Help Forum Discussions

Finding limit of this function, using Limit rules

Find ∫ 1/(1-4x^2) dx with a low limit of 0 and a high limit of 1

limit question on a limit comparison test problem.

[SOLVED] limit of function exists, but limit of its derivative doesn't.

Limit, Limit Superior, and Limit Inferior of a function