202111221624 Exercises 9.1.A (Q2)

For Exercises 1-8, determine if the given sequence is convergent. If so then find its limit.

2. $\displaystyle{\bigg\{ \frac{n^2}{3n^2+7n-2}\bigg\}_{n=1}^{\infty} }$

Solution.

By L’Hôpital’s Rule, treating an integer $n \ge 1$ as a real-valued variable $x$ ,

$\begin{aligned} &\quad \lim_{n\to\infty}\frac{n^2}{3n^2+7n-2} \\ & = \lim_{n\to\infty}\frac{2n}{6n+7} \\ & = \frac{2}{6} \\ & = \frac{1}{3}\\ \end{aligned}$

Thus the sequence is convergent and its limit is $1/3$ .

M	T	W	T	F	S	S
1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30

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