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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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This is true IF both limits that you just mentioned exist. –  Graphth Nov 6 '11 at 14:39
    
@Graphth Of course. –  Paul Manta Nov 6 '11 at 14:39
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Yes, as long as you don't hit an indeterminate form. –  J. M. Nov 6 '11 at 14:39
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See proofwiki.org/wiki/Combination_Theorem_for_Sequences/… for the proof. –  Martin Sleziak Nov 6 '11 at 14:42

2 Answers 2

up vote 5 down vote accepted

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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