# Does limit ln(n)/n^c -> 0 for any c>0?

• grossgermany
In summary, the conversation discusses the question of whether the limit ln(n)/n^c approaches 0 for any c>0. The conversation also mentions different methods for proving this statement, such as using epsilon-delta proofs or BigO SmallO notation. The conversation concludes by mentioning the usefulness of the Bolzano Weierstrasse Theorem in approaching this problem in real analysis.
grossgermany

## Homework Statement

Does limit ln(n)/n^c -> 0 for any c>0?

## The Attempt at a Solution

I wonder if there is an
1.Epsilon Delta Proof
2.Proof using BigO SmallO notation.

Thanks

It is obviously true. There are a lot of ways to prove this. What kind of class is this for? Are you expected to be rigorous in your solutions? N-epsilon proofs and asymptotic behaviour are very different arguments in terms of the development required in an analysis setting.

Yes, this is for real analysis class

What tools do you have available? How was ln(n) derived?

There is a nifty theorem that says that if a sequence converges then any subsequence will converge to the same limit. This will enable you to look at ln(x)/x^c in R rather than in N.

Yes, we can use Bolzano Weierstrasse Theorem. Please feel free to proceed.
My level is on baby Rudin, the this is the first course in real analysis.

## 1. What is the significance of the limit ln(n)/n^c?

The limit ln(n)/n^c represents the growth rate of the natural logarithm compared to a polynomial function. It is commonly used in the analysis of algorithms and in studying the rate of growth of a sequence.

## 2. Is the limit ln(n)/n^c always equal to 0?

No, the limit ln(n)/n^c is only equal to 0 for values of c greater than 1. For values of c less than or equal to 1, the limit approaches infinity as n approaches infinity.

## 3. Can the limit ln(n)/n^c be evaluated for negative values of n?

No, the natural logarithm ln(n) is only defined for positive values of n. Therefore, the limit ln(n)/n^c cannot be evaluated for negative values of n.

## 4. How is the limit ln(n)/n^c affected by changes in the value of c?

The limit ln(n)/n^c is inversely proportional to the value of c. This means that as c increases, the limit decreases and approaches 0. As c decreases, the limit increases and approaches infinity.

## 5. Is there a relationship between the limit ln(n)/n^c and the natural logarithm ln(n)?

Yes, the limit ln(n)/n^c is equivalent to the natural logarithm ln(n) when c=1. This is because ln(n)/n^1 reduces to ln(n). However, for any other value of c, the limit ln(n)/n^c is a different function with a different rate of growth.

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