Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function. R(x)= (5x^2-19x-4)/ 2x^2-7x-4
Find the horizontal asymptotes.
To find the vertical asymptotes, set the denominator of the rational function equal to zero and solve for x:
2x^2 - 7x - 4 = 0
This quadratic equation can be factored as:
(2x + 1)(x - 4) = 0
Setting each factor equal to zero gives us the solutions x = -1/2 and x = 4.
Therefore, the vertical asymptotes are x = -1/2 and x = 4.
To find the horizontal asymptotes, we compare the degrees of the numerator and denominator of the rational function. Since both the numerator and denominator have the same degree of 2, we look at the leading coefficients of both:
Leading coefficient of the numerator: 5
Leading coefficient of the denominator: 2
To find the horizontal asymptote, divide the leading coefficient of the numerator by the leading coefficient of the denominator:
5/2 = 2.5
Therefore, the horizontal asymptote is y = 2.5.
There are no oblique asymptotes for this rational function.