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.

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Evaluate limn→−∞an\(\lim\)_{n\(\to\)-\(\infty\)}a_{n}limn→−∞an and determine whether the sequence converges or diverges.an=300sin(n2−120n)a_{n}=300\(\sin\[\left\)(\(\frac{n^2-120}{n}\]\right\))an=300sin(nn2−120)

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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\))$