Use of Hôpital's rules to calculate also the sequences 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.
 A: The analog of L'Hospital rule for sequences is the Stolz-Cesaro theorem:
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwjAvLyEgYnrAhUlJTQIHayoC_YQFjAAegQIAxAB&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FStolz%25E2%2580%2593Ces%25C3%25A0ro_theorem&usg=AOvVaw1iQkM4oA1MZoySX8fqNTXe
A: 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.
