Proving Limit at Infinity: n^(1/n) = 1

[IniquiTrance]

Homework Statement

How can I prove that:

$$\lim_{n \rightarrow \infty} n^{\frac{1}{n}}=1$$

Isn't $$\infty^{0}$$ indeterminate?
Thanks!

Homework Equations

The Attempt at a Solution

 

[Dick]

Yes, it is indeterminant. That's not the end of the story. Indeterminant just means you don't know what the limit is yet. Take the log. Can you prove (1/n)*log(n) approaches 0?

 

[IniquiTrance]

That becomes $$0*\infty$$, isn't that indeterminate as well?

 

[w3390]

Can't you use L'Hopital's rule?

 

[IniquiTrance]

Right. Thanks guys!