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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 $l_b \ne 0$?

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Yes, if both limits exist and the denominator is nonzero, then this is correct (and usually by far the easiest way to find the limit).

Ultimately this is because the function $(x,y)\mapsto \frac xy$ is continuous for $y\ne 0$.

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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.

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