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For the following sequence, how do I find if it converges and if so how do I find its limits.

$$a_n = \frac{12−8n}{4n+36}, (n=1,2,3,...)$$

What are the steps that I need to follow to get the answer?

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Hint: Divide numerator and denominator by $n$. Then check what the limit is as $ n$ tends to $\infty$.

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There are several possibilities to solve this problem.

One was already mentioned by lsp.

Divide numerator and denominator by n (the limit should stay the same) and check wath happens when $n$ goes to infinity. Remember ($\frac{a}{n} \to 0$ when $n \to \infty$)

You dan also see the limit as the asymptotic limit of two continuous functions (make the $n$ continuous instead of discrete. This subsequence should converge to the same limit.) With de l'Hospital: $$ \lim_{n \to \infty} \frac{12-8n}{4n+36} = \lim_{n \to \infty} \frac{-8}{4} = -2 $$

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