Determine the convergence or divergence of the sequence with the given \( n \)th term. If the sequence converges, find its limit. (If the quantity diverges, enter DIVERGES.) \[ a_{n}=\frac{6}{n+9} \]

To determine whether the sequence \( a_n = \frac{6}{n+9} \) converges or diverges, and to find its limit if it converges, we analyze its behavior as \( n \) approaches infinity. 1. **Examine the Denominator:** - As \( n \) increases, \( n + 9 \) also increases without bound. Mathematically, \(\lim_{n \to \infty} (n + 9) = \infty\). 2. **Evaluate the Sequence:** - Since the denominator grows without bound, the overall fraction \( \frac{6}{n+9} \) becomes smaller and smaller. - Mathematically, \(\lim_{n \to \infty} \frac{6}{n+9} = 0\). 3. **Conclusion:** - The sequence converges because it approaches a specific finite limit. - The limit of the sequence is **0**. **Answer:** The sequence converges and its limit is 0