Limit of zero to the zero

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

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

EDIT: OK solved.

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

Re: Limit of zero to the zero

Quote:

Originally Posted by

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

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

EDIT: OK solved.

$\displaystyle \lim\limits_{n \to \infty} \ln \left[\left(\frac{1}{n}\right)^\frac{1}{n}\right]$ $\displaystyle = \lim\limits_{n \to \infty} \frac{\ln \frac{1}{n}}{n}$ 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 $\displaystyle 0^0 = 0$. You have done the limit correctly (as far as I can see) but the problem $\displaystyle 0^0$ has been labelled as undefined.

-Dan

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"

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