Hoping that my question is clear, I would like to understand because the L'Hôpital's rules are used in several questions on Math.SE (an answer for example) to calculate the sequences,
$$(a_n)_{n\in\Bbb N}, \quad \text {or} \quad \{a_n\}, \quad n\in\Bbb N$$
During my university period I had been instructed that Hôpital's theorems cannot be applied.
In this answer, L'Hôpital's rule is actually used to compute $$ \lim_{\substack{x \to \infty \\ x \in \Bbb R}} \frac{ \ln(1-\frac{3}{x})}{1/x} = \lim_{\substack{x \to \infty \\ x \in \Bbb R}} \frac{\frac{3}{x^2}}{\frac{-1}{x^2} (1-\frac{3}{x})} = -3 \, , $$ and that implies $$ \lim_{\substack{n \to \infty \\ n \in \Bbb N}} \frac{ \ln(1-\frac{3}{n})}{1/n} = -3 \, . $$ Only in that answer $n$ is used both as the (integer) index of the sequence and as the real-valued argument of a function.
Generally, if your sequence is $a_n = f(n)$ with a function $f: [1, \infty) \to \Bbb R$, and if you can show that $\lim_{x \to \infty} f(x) = A$ (using L'Hôpital's rule or any other method), then $\lim_{n \to \infty} a_n = A$ follows.