What are the horizontal and vertical asymptotes of the function f(x) = (3x² + 2x + 13)/(6x² - 5x - 6)?

To determine the horizontal and vertical asymptotes of the function ( f(x) = \frac{3x^2 + 2x + 13}{6x^2 - 5x - 6} ), we analyze the degrees of the polynomials in both the numerator and the denominator.

For horizontal asymptotes, when the degrees of the numerator and denominator are equal—both are degree 2 in this case—the horizontal asymptote can be found by taking the ratio of the leading coefficients. The leading coefficient of the numerator ( (3x^2) ) is 3, and for the denominator ( (6x^2) ) is 6. Thus, the horizontal asymptote is given by:

[

y = \frac{3}{6} = \frac{1}{2}

]

This confirms that the horizontal asymptote is ( y = \frac{1}{2} ).

Next, we find the vertical asymptotes by setting the denominator equal to zero and solving for ( x ). The denominator ( 6x^2 - 5x - 6 = 0 ) can be solved using the quadratic formula ( x = \frac{-b