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 oblique asymptotes.
To find the vertical asymptotes, set the denominator 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:
2x + 1 = 0 or x - 4 = 0
x = -1/2 x = 4
Therefore, the vertical asymptotes are x = -1/2 and x = 4.
To find the horizontal asymptote, compare the degrees of the numerator and denominator. Since they are both quadratic terms, divide the leading coefficients to find the horizontal asymptote:
Horizontal asymptote: y = 5/2
To find the oblique asymptote, perform long division of the numerator by the denominator:
5x - 4
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2x^2 - 7x - 4 | 5x^2 - 19x - 4
- (5x^2 - 17x + 8)
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-2x - 12
This gives a quotient of 5 with a remainder of (-2x - 12)/(2x^2 - 7x - 4).
Thus, the oblique asymptote is y = 5 with a slant of -2x - 12 over the quadratic term.