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So, I found lots of exercises for $\lim_{n \to +\infty}$ with $\sqrt[n]{\ln(n)}$ in my calculus book and I didn't know how to solve those. The example is $\lim_{n \to +\infty} n(1 + \sqrt[n]{\ln(n)})$.

I actually know the limit from this post: Limit:$ \lim\limits_{n\rightarrow\infty}\left ( n\bigl(1-\sqrt[n]{\ln(n)} \bigr) \right )$. The problem is, all answers use "Taylor, h'Hospital or d'Alembert" and I didn't learn about those theorems yet. So my question is - is there any way to deal with $\lim_{n \to +\infty} \sqrt[n]{\ln(n)}$ (taking $\lim_{n \to +\infty} n(1 + \sqrt[n]{\ln(n)})$ as an example) using most basic calculus?

By most basic calculus I mean not using Taylor, h'Hospital nor d'Alembert.

Aryaman Maithani
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theboyboy
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1 Answers1

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Here is a hint:

For $n\geq e^2$

$$ e^{1/n}<(\log n)^{1/n}\leq n^{1/n} $$

There are several "elementary" methods to show that

  1. $\lim_{n\rightarrow\infty}p^{1/n}=1$ for $p>0$ constant and
  2. $\lim_{n\rightarrow\infty}n^{1/n}=1$ See for instance Rudin's Principle's of mathematical analysis, p. 57.

The conclusion follows from the squeeze lemma.

Proofs for 1 and 2 type of methods also have been explained on this site.

Mittens
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