# Is this a correct 'trick' to find the limit of a sequence?

Let's say I'm given a sequence of the form $c_n = \frac{a_n}{b_n}$ and I'm asked to find its limit but I don't know how to do it directly. Is it correct for me to say that its limit is $\frac{l_a}{l_b}$, where $l_a = \lim a_n$ and $l_b = \lim b_n$, with $l_b \ne 0$?

Ultimately this is because the function $(x,y)\mapsto \frac xy$ is continuous for $y\ne 0$.
This is true IF both limits that you just mentioned exist. An easy example is $a_n = b_n = n$. Then, $c_n = \frac{n}{n}$, which has limit $1$, but $\lim\limits a_n = \lim\limits b_n = \infty$. In the case of such an indeterminate form, then l'Hopital's rule might be your best tool.