Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3ⁿ⁺¹ + 3)⁄3ⁿ}
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14. Sequences & Series
Sequences
2:15 minutesProblem 10.2.51
Textbook Question
Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{nsin(6 / n)}
Verified step by step guidance1
Identify the sequence given: \(a_n = n \sin\left(\frac{6}{n}\right)\).
Recall the limit property for sine near zero: \(\lim_{x \to 0} \frac{\sin x}{x} = 1\).
Rewrite the sequence term to use this property by setting \(x = \frac{6}{n}\), so \(a_n = n \sin\left(\frac{6}{n}\right) = \frac{\sin\left(\frac{6}{n}\right)}{\frac{6}{n}} \times 6\).
Analyze the limit as \(n \to \infty\): since \(\frac{6}{n} \to 0\), use the limit property to find \(\lim_{n \to \infty} \frac{\sin\left(\frac{6}{n}\right)}{\frac{6}{n}} = 1\).
Combine the results to conclude that \(\lim_{n \to \infty} a_n = 6 \times 1 = 6\).
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Sequence
The limit of a sequence is 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.
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Introduction to Sequences
Behavior of the Sine Function for Small Arguments
For values of x close to zero, sin(x) is approximately equal to x. This linear approximation, sin(x) ≈ x, is useful for evaluating limits involving sine functions where the argument tends to zero.
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Limit Laws and Substitution
Limit laws allow the evaluation of limits by breaking complex expressions into simpler parts. Substitution involves replacing variables with their limiting values when the function is continuous, facilitating the calculation of the sequence's limit.
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