23. Intro to Derivatives & Area Under the Curve

Limits of Sequences

23. Intro to Derivatives & Area Under the Curve

Limits of Sequences

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Multiple Choice

Evaluate $\lim_{n\to-\infty}a_{n}$ and determine whether the sequence converges or diverges.
$a_{n}=300\sin\left(\frac{n^2-120}{n}\right)$

$a_{n}=300$ , converges.

$a_{n}=300$ , diverges.

$a_{n}=DNE$ , converges.

$a_{n}=DNE$ , diverges.

Verified step by step guidance

First, identify the sequence given: a_n = 300\sin\left(\frac{n^2 - 120}{n}\right).

To evaluate the limit as n approaches negative infinity, consider the expression inside the sine function: \frac{n^2 - 120}{n}.

Simplify \frac{n^2 - 120}{n} to n - \frac{120}{n}. As n approaches negative infinity, \frac{120}{n} approaches 0, so the expression simplifies to n.

The sine function, \sin(n), oscillates between -1 and 1 for all real numbers n. Therefore, \sin(n) does not settle to a single value as n approaches negative infinity.

Since \sin(n) oscillates indefinitely, the sequence a_n = 300\sin(n) does not converge to a single value, and thus the limit does not exist (DNE), indicating divergence.