12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)

검증된 단계별 안내

Identify the given sequence: $a_n = \frac{(3n^2 + 2n + 1) \cdot \sin(n)}{4n^3 + n}$.

Analyze the behavior of the numerator and denominator separately as $n$ approaches infinity. The numerator grows roughly like \$3n^2$ times \(\sin(n)$, and the denominator grows like \)4n^3$.

Note that $\sin(n)$ oscillates between $-1$ and $1$, so $-1 \leq \sin(n) \leq 1$. Use this to create inequalities for $a_n$:

\[-\frac{3n^2 + 2n + 1}{4n^3 + n} \leq a_n \leq \frac{3n^2 + 2n + 1}{4n^3 + n}.\]

Simplify the bounding expressions by dividing numerator and denominator by $n^3$ to understand their limits:

\[-\frac{3/n + 2/n^2 + 1/n^3}{4 + 1/n^2} \leq a_n \leq \frac{3/n + 2/n^2 + 1/n^3}{4 + 1/n^2}.\]

Apply the Squeeze Theorem: since both bounds approach $0$ as $n \to \infty$, conclude that $\lim_{n \to \infty} a_n = 0$.

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주요 개념

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Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Squeeze Theorem

The Squeeze Theorem helps find limits by 'trapping' a sequence between two others that have the same limit. If aₙ ≤ bₙ ≤ cₙ for all n beyond some point, and both aₙ and cₙ converge to L, then bₙ also converges to L.

Behavior of Trigonometric Functions in Limits

Trigonometric functions like sin(n) oscillate between -1 and 1 and do not have limits as n approaches infinity. When combined with sequences that tend to zero, their bounded nature allows the use of the Squeeze Theorem to evaluate the overall limit.

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)