Get the answer to "Find the limits of the sequences 1){n2+3n3+n2−1}n=1∞ 2){nsin⁡πn}n=1∞"

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Marvin Mccormick

Answered question

2021-08-21

Find the limits of the sequences
1) ${\frac{n^{2} + 3}{n^{3} + n^{2} - 1}}_{n = 1}^{\infty}$
2) ${n \sin \frac{π}{n}}_{n = 1}^{\infty}$

Answer & Explanation

lamanocornudaW

Skilled2021-08-22Added 85 answers

1)
$lim_{n \to \infty} \frac{n^{2} + 3}{n^{3} + n^{2} - 1} = lim_{n \to \infty} \frac{n^{2} (1 + \frac{3}{n^{2}})}{n^{3} (1 + \frac{1}{n} - \frac{1}{n^{3}})}$
$lim_{n \to \infty} \frac{n^{2} + 3}{n^{3} + n^{2} - 1} = lim_{n \to \infty} \frac{(1 + \frac{3}{n^{2}})}{n (1 + \frac{1}{n} - \frac{1}{n^{3}})} = 0$
$lim \frac{n^{2} + 3}{n^{3} + n^{2} - 1} = 0$
2)
$lim_{n \to \infty} n \sin \frac{π}{n} = lim_{n \to \infty} \frac{\sin \frac{π}{n}}{\frac{1}{n}} = lim_{n \to \infty} \frac{π (\sin \frac{π}{n})}{(\frac{π}{n})}$
$= lim_{n \to \infty} \frac{π \sin \frac{π}{n}}{(\frac{π}{n})}$

Since $lim_{n \to \infty} (\frac{\sin \frac{1}{n}}{\frac{1}{n}}) = lim_{n \to 0} \frac{\sin n}{n} = 1$
$lim_{n \to \infty} n \sin \frac{π}{n} = π$

Jeffrey Jordon

Expert2021-10-23Added 2605 answers

Answer is given below (on video)

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