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Suppose I have 2 seq, $a_n, b_n>0\forall n$ and a, b their limit respectively. If $c_n=\frac{a_n}{b_n}$ then it will converge if $b>0$ right? Is their any other condition for convergence of $c_n$?

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Yes, that is true. –  M.B. Jun 2 '12 at 12:25
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Notice by the limit laws we have:

$$\lim_{n\to\infty}c_n=\lim_{n\to\infty}\frac{a_n}{b_n}=\frac{\lim_{n\to\infty}a_n}{\lim_{n\to\infty}b_n}=\frac{a}{b}$$

if and only if $b\ne 0$.

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