Question

suppose that \\(\lim_{n \to \infty} a_n = 1\\), \\(\lim_{n \to \infty} b_n = -1\\), and \\(0 < -b_n < a_n\\) for all \\(n\\). evaluate the limit. (if an answer does not exist, enter dne. if there is not enough information to determine whether the limit exists, enter not enough information.)

\\\lim_{n \to \infty} 5a_n - 2b_n\\

Explanation:

Identify given limits and target expression

Given:

$$ \lim_{n \to \infty} a_n = 1,\quad \lim_{n \to \infty} b_n = -1 $$

Target:

$$ \lim_{n \to \infty} (5a_n - 2b_n) $$

Apply limit laws for sequences

$$
LATEXBLOCK0
$$

Compute the final value

$$ 5(1) - 2(-1) = 5 + 2 = 7 $$

Answer:

Suppose that \(\lim_{n \to \infty} a_n = 1\), \(\lim_{n \to \infty} b_n = -1\), and \(0 < -b_n < a_n\) for all \(n\). Evaluate the limit. (If an answer does not exist, enter DNE. If there is not enough information to determine whether the limit exists, enter NOT ENOUGH INFORMATION.)

$$\lim_{n \to \infty} 5a_n - 2b_n$$

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