For each natural number $n \geq 1$, let $a_n = \frac{n}{10^[\log_{10}n]}$ , where $[x]$ = smallest integer greater than or equal to $x$.

Question

For each natural number $n \geq 1$, let $a_n = \frac{n}{10^{[\log_{10}n]}}$ , where $[x]$ = smallest integer greater than or equal to $x$. Which of the following statements are true?

1. $\liminf a_n = 0$

2. $\liminf a_n$ does not exist

3. $\liminf a_n = 0.15$

4. $\limsup a_n = 1$

If I take $n = {10}^n$ then $a_n = \frac{{10}^n}{10^{[\log_{10}{10}^n]}} = 1$, $\forall n \in \Bbb N$ So $\limsup a_n = 1$ but I am not sure about it. I have no information how discard other options.