$\left\{ a_n\right\} $, does it converge?
$$\lim_{x \to \infty}a_n\ = \frac{f(n)}{g(n)} \Rightarrow \frac{0}{0} form $$
$$\lim_{x \to \infty}a_n\ = \frac{f'(n)}{g'(n)}, LHR$$
How can this assumption be made to a sequence that uses integers, without sometimes causing significant effects, for instance:
$\lim_{n \to \infty}sin(\pi n)DNE $, if n is continuous. However,
$\lim_{n \to \infty}sin(\pi n)= 0$, if n is confined to integers.
When can this assumption be made? How can one tell if the function is not as obvious as the above examples?