Question

Describe the behavior of f(n)=n^(1/n) in words.

Answer 1

End behavior? or as n approaches 0? or both?

Answer 2

In any case, you'll have to take the limit:
\[\large \lim_{n\to c}f(n)=\cdots\]
where you choose \(c=0,\pm\infty\).

Answer 3

My paper says as n grows large

Answer 4

End behavior it is!

Need help finding the limits? :
\[\lim_{n\to\infty}n^{1/n}\]
\[\lim_{n\to-\infty}n^{1/n}\]

Answer 5

The reason I can't figure it out is because at one point in an example question, it says f(n) approaches a limit of 2.7 because the graph continually increases, but never gets equal to or greater than 2.7. The graph f(n)=\[n ^{1/n}\] increases to a certain point and then decreases, so what am I supposed to say?

Answer 6

Well it sounds like, in addition to finding the limit at infinity, you also have to find the local maximum for positive \(n\). Is that right?

To find the limit: do you know L'Hopital's rule?

Answer 7

I don't think so:( I took precalculus last year and this is the assignment to start us into calculus. What is L'Hopital's rule?

Answer 8

You'll learn soon enough. It's a fairly easy way to determine limits at 0 and ±infinity.

To find this limit, I think it would suffice to show what happens for certain values of \(n\). For example, see the following table:
The value of \(n^{1/n}\) increases for a while then dips back down for some \(n\) between 2 and 4.

To find out exactly when the dip occurs, you need to use the first derivative test. Have you learned that yet?