Find the limit of the following sequence or determine that the sequence diverges. \( \left.\qquad \frac{\sin 9 n}{4 n}\right\} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The limit of the sequence is (Type an exact answer.) B. The sequence diverges.
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To find the limit of the sequence \( \frac{\sin(9n)}{4n} \), we can analyze its behavior as \( n \) approaches infinity. We know that the sine function oscillates between -1 and 1. Therefore, we can establish the following bounds: \[ -\frac{1}{4n} \leq \frac{\sin(9n)}{4n} \leq \frac{1}{4n}. \] As \( n \) approaches infinity, both \( -\frac{1}{4n} \) and \( \frac{1}{4n} \) converge to 0. By the Squeeze Theorem, since \( \frac{\sin(9n)}{4n} \) is squeezed between two bounds that both converge to 0, we conclude that: \[ \lim_{n \to \infty} \frac{\sin(9n)}{4n} = 0. \] Thus, the correct choice is: A. The limit of the sequence is 0.
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