Find the limit of the following sequence or determine that the sequence diverges. \[ \left\{\frac{2 n^{2}-1}{8 n^{2}+1}\right\} \]

To find the limit of the sequence \[ a_n = \frac{2n^2 - 1}{8n^2 + 1} \] as \( n \) approaches infinity, we can start by dividing both the numerator and the denominator by \( n^2 \): \[ a_n = \frac{2 - \frac{1}{n^2}}{8 + \frac{1}{n^2}}. \] As \( n \) approaches infinity, the terms \( \frac{1}{n^2} \) in both the numerator and the denominator approach \( 0 \). Thus, we get: \[ \lim_{n \to \infty} a_n = \frac{2 - 0}{8 + 0} = \frac{2}{8} = \frac{1}{4}. \] Therefore, the limit of the sequence is \[ \boxed{\frac{1}{4}}. \]