Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
a.If limₙ→∞aₙ = 1 and limₙ→∞bₙ = 3, then limₙ→∞(bₙ / aₙ) = 3.

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Recall the limit laws for sequences: if \( \lim_{n \to \infty} a_n = A \) and \( \lim_{n \to \infty} b_n = B \), and if \( A \neq 0 \), then \( \lim_{n \to \infty} \frac{b_n}{a_n} = \frac{B}{A} \).

Given \( \lim_{n \to \infty} a_n = 1 \) and \( \lim_{n \to \infty} b_n = 3 \), since \( a_n \) approaches 1 (which is not zero), the limit of the quotient should be \( \frac{3}{1} = 3 \).

Therefore, the statement \( \lim_{n \to \infty} \frac{b_n}{a_n} = 3 \) is true under these conditions.

To confirm, consider that the denominator sequence \( a_n \) does not approach zero, so division by \( a_n \) is well-defined in the limit.

Hence, the limit of the quotient is the quotient of the limits, which justifies the statement.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. If a sequence converges to a limit, its terms get arbitrarily close to that limit for sufficiently large indices.

Limit Laws for Sequences

Limit laws allow us to compute limits of combined sequences using operations like addition, multiplication, and division, provided the limits of individual sequences exist and, in the case of division, the denominator's limit is not zero.

Counterexamples in Limit Problems

Counterexamples demonstrate that a general statement is false by providing a specific case where the statement fails. In limit problems, they help test the validity of limit laws or assumptions, especially when conditions like nonzero denominators are involved.

Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.a.If limₙ→∞aₙ = 1 and limₙ→∞bₙ = 3, then limₙ→∞(bₙ / aₙ) = 3.