Limit Proof – Simple Proof of $\lim_{n\to \infty} \frac{\ln n}{n}=0$ for High School Students

Question

The question refers to the mathematics course for the students of a fifth scientific high school, whereas the order of the arguments of the textbook is almost identical to what I treated when I was studying at the university.

Here a graph of the arguments:

$$\color{brown}{\text{sequences}}\to \color{red}{\text{topology in }\Bbb R \text{ and }\Bbb R^2}\to\color{gray}{\text{limits of functions}}\to$$
$$\color{magenta}{\text{continuity}}\to \color{cyan}{\text{discontinuity}}\to\color{teal}{\text{derivates}}\to$$
$$\color{blue}{\text{max and min}}\to \color{green}{\text{study of functions in reals}}\to\color{orange}{\text{indefinite and definite integration}}$$
etc.

We suppose that we have this limit $$\lim_{n\to \infty} \frac{\ln n}{n}$$

it goes to $0$, because for $n\in \Bbb N$ large, I have $0\leq \ln n<n$, i.e. the natural logarithm sequence of $n$, i.e. $\{\ln n\}$ is much slower than the sequence $\{n\}$. i.e. we say that the sequence $\{n\}$ is predominant to $\{\ln n\}$; hence it is "similar" to have

$$\bbox[yellow,5px]{\lim_{n\to \infty} \frac{\text{constant}}{n}=0}$$

Is there an alternative clear proof that $$\lim_{n\to \infty} \frac{\ln n}{n}=0, \quad ?$$

Is there also something like:
$$\bbox[orange,5px]{\lim_{n\to \infty} \frac{\ln(f(n))}{g(n)}}$$

Is there any known limit if $f(n)$ and $g(n)$ are two polynomials with $$\deg(f(n))\gtreqless\deg(g(n))\quad ?$$

---

Answer 1

The standard proof I know for high-school consists in proving first (with derivatives) that $\;\ln x < \sqrt x\;$ for all $x>0$.

Then one deduces that, for all $n\ge 1$, $$0\le\frac{\ln n}n<\frac{\sqrt n}n=\frac1{\sqrt n},$$ and observes the latter expression tends to $0$.

For the last question, as far as I know, requires asymptotic analysis, namely finding an asymptotic equivalent for $f(n)$ and $g(n)$ (their leading monomials), but I don't think this is in the high school cursus in any country.

Edit: Actually, one may circumvent the use of asymptotic analysis,at the cost of a slightly longer proof. Let: \begin{align}f(n)&= an^k + \sum_{i=k-1}^0 a_ix^i,& g(n)&= cn^l + \sum_{i=l-1}^0 c_ix^i. \end{align} Then we may write $\ln (f(n)=\ln(an^k)+\sum_{i=k-1}^0\frac{a_i}{a x^{k-i}}$, so that $$\frac{\ln\bigl(f(n)\bigr)}{g(n)}=\frac{\ln\ a}{g(n)}+\frac{k\ln(n)}{g(n)},$$ and it is a simple routine to show that each of these fractions tends to $0$ when $n$ tends to $\infty$.