12. Find the horizontal and vertical asymptotes. (Do not sketch) \( f(x)=\frac{-2 x^{2}+4 x+1}{9 x^{2}-1} \)

To find the horizontal asymptote of the function \( f(x) = \frac{-2x^2 + 4x + 1}{9x^2 - 1} \), we look at the leading coefficients of the polynomial in both the numerator and denominator. Both are second-degree polynomials (highest power of x terms), so we take the ratio of their leading coefficients: \( \frac{-2}{9} \). Thus, the horizontal asymptote is \( y = -\frac{2}{9} \). For the vertical asymptotes, we set the denominator equal to zero: \( 9x^2 - 1 = 0 \). Solving for \( x \) gives \( 9x^2 = 1 \) or \( x^2 = \frac{1}{9} \). This results in \( x = \frac{1}{3} \) and \( x = -\frac{1}{3} \). Therefore, the vertical asymptotes are located at \( x = \frac{1}{3} \) and \( x = -\frac{1}{3} \).