Finding the limit of $\frac{N_n}{\ln(n)}$ where $N_n$ is the number of digits of $n$

Question

I came across this question in an entrance exam of a local college where we are asked to evaluate the limit : $$\lim\limits_{n\to\infty} \frac{N_n}{\ln(n)}$$ Where $N_n$ denotes the number of digits of $n$, with the latter being a non zero positive integer. It seems I am lacking some sort of relationship between $N_n$ and $n$ (i.e equality/inequality). I could spot that $n \ge N_n$ but it doesn't seem useful in this case. Appreciate any help!

Answer 1

HINT

Following the suggestion given in the comments, we have that

$$\frac{\log_{10}n}{\ln(n)}\le \frac{N_n}{\ln(n)}\le \frac{1+\log_{10}n}{\ln(n)}$$

then we can conclude by squeeze theorem.